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University Microfilms International 300 N /EE B ROAD. ANN ARBOR, Ml 48106 10 BEDEORD ROW. LONDON WC 1 R 4EJ, ENGLAND 8001817

Ro t h , Ro b er t Lyle, j r .

HALL TRIPLE SYSTEMS AND COMMUTATIVE MOUFANG EXPONENT 3 LOOPS

The Ohio State University PH.D. 1979

University Microfilms International 300 N. Zeeb Road, Aim Aitooi, MI 48106 18 Bedford Row, London WC1R 4EJ. England HALL TRIPLE SYSTEMS AND COMMUTATIVE MOUFANG

EXPONENT 3 LOOPS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Robert Lyle Roth, Jr., B.S., M.Sc,

*****

The Ohio State University

1979

Reading Committee; Approved By

Professor D. K. Ray-Chaudhuri Professor E. Bannai Professor T. A. Dowling ^ i ~* Professor G. N. Robertson Professor J. C. D. S. Yaqub Adviser Department of Mathematics This work is dedicated to my mother and to the memory of my father. Their love will be my treasure always.

il ACKNOWLEDGEMENTS

The subject of this dissertation was suggested by my

adviser, Professor D. K. Ray-Chaudhuri. I wish to express my

gratitude for his helpful suggestions, his motivating questions,

and his encouragement in the preparation of this work, in

addition I want to thank him for the many excellent courses and

seminars he has presented - they have given me the chance to see

and learn some of the beauty of combinatorial mathematics. Part

of Professor Ray-Chaudhuri’s time spent on discussion and reading

related to this dissertation was supported by NSF grant number

MCS75-08231-A01.

I also want to thank Professors J. C. D. S. Yaqub, A. P.

Sprague, and J. Ferrar for their kindness in discussing various

aspects of this work with me.

I am grateful to professors T. A. Dowling, R. M. Wilson,

E. Bannai, G. N* Robertson, and R. p. Gupta for the training they have provided. I also want to acknowledge these excellent teachers who have shared a part of their knowledge with m e : Professors

J. P. Tull, R. Gold, and A. Ross at The Ohio State University and

Professors N. C. Ankeny, J. Munkres, G. B. Thomas, A. P. Mattuck,

E. Speer, and G. Sacks at the Massachusetts Institute of Technology.

ill My appreciation goes to Professors D. Kelly and M. Splvak who sparked my first interest in becoming a mathematician.

I am indebted to Mrs. D. Shapiro and Mrs. M. Greene for their excellent technical assistance in the preparation of this treatise.

I want to express my heartfelt thanks to my wife-to-be

Kathy for her kindness, patience, and loving support.

iv VITA

May 8, 1952...... Born - Jacksonville, Florida

197^ ...... B.S., Massachusetts Institute of Technology, Cambridge, Massachusetts

1 9 7 ^ - 1 9 7 5 ...... University Fellow, Department of Mathematics, The Ohio State University, Columbus, Ohio

1975-1977...... Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio

1976 • * ...... M.Sc., The Ohio State University, Columbus, Ohio

1977-197 8...... University Fellow, The Ohio State University, Columbus, Ohio

1978-197 9...... Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio

FIELD OF STUDY

Major Field: Mathematics

Studies in Combinatorial Theory. Professor D. K. Ray-Chaudhuri

v TABLE OF CONTENTS

Page DEDICATION...... II

ACKNOWLEDGEMENTS ...... iii

VITA ...... v

LIST OF T A B L E S ...... viii

LIST OF F I G U R E S ...... ix

INTRODUCTION ...... 1

Chapter

I. PRELIMINARIES; HISTORY AND DEFINITIONS...... 6

Conventions and Basics Hall's Original Work Characterization of Affine Spaces Perfect Matroid Designs Coordinatization by Loops HTS’s in New Contexts

II. AN ACCOUNT OF BRUCK'S THEORY OF MOUFANG LOOPS . . . 79

Moufang Loops Commutative Moufang Loops New Contributions to the Theory

III. THE BRUCK CONSTRUCTION...... ldA

The Loops Bd and The Malbos Variation

IV. COMMUTATIVE MOUFANG EXPONENT 3 LOOPS OF NILFOTENCE CLASS 2 ...... 135

The Behavior of the Product The Free Construction The Construction of Loops with Specified Properties

vi Page V. THE CATALOGUE OF HALL TRIPLE SYSTEMS OF CARDINALITY AT MOST ^ , ...... I78

• 1 li General Remarks and the Cases |G| - 3 and IGI-35 The Case |g | = 3

VI. COMMUTATIVE MOUFANG EXPONENT 3 LOOPS OF NILPOTENCE CLASS 3 ...... 203

The Behavior of the Product The Construction of F .

VII. MISCELLANEOUS RESULTS...... 236

Designs Derived from HTS's Automorphism Groups Concluding Remarks

BIBLIOGRAPHY...... 261+

vii LIST OF TABIES

Page Table

1. R(m1,m2) ...... Ill

2. Dependencies among the Generating Basic 2-Associators in a c-SPf exp-3 Loop of Dimension ^ and Nilpotence Class 3 ...... 189

3 . Equivalence of 6- t u p l e s ...... 195

U. Equivalence of 4-tuples ...... 201

5 . The Blocks of Dfi(H8l) ...... 251

viii LIST OF FIGURES

Page Figure

1. Pasch A x i o m ...... 25

2. The Loop Based at e ...... 'jh

3* The Loop of a 3 - N e t ...... 58

Reflection of 1-Lines...... 59

5 . Reflection and the C o r e ...... 90

ix INTRODUCTION

The purpose of this dissertation is to study a class of * Steiner Triple Systems called Hall Triple Systems (abbreviated

HTS*s) which are very special with regard to a geometric property

they possess.

The sets of axioms for affine spaces that have been used

include a "three-dimensional" axiom. Fur example, in the

axiomatization due to Lenz [37] it is the transitivity of parallelism

and in the axiomatization used by Sasaki in [53] it is the require­ ment that a non-empty intersection of two distinct planes contained

in a 3-flat must be a line. It is natural to ask if such an axiom

can be replaced by a "two-dimensional" axiom so that the resulting

set of axioms is equivalent to the original one. The only good

candidate for such an axiom is the requirement that every plane be an affine plane. (Note that following immediately from the classical axioms of Veblen and Young for projective spaces is the fact that any linear space containing three non-collinear points and having the property that every plane is a projective plane must be a projective space.) in a very beautiful work [12]

Buekenhout has shown that this "two-dimensional" axiom can replace

* A H the necessary definitions which we take for granted here will be stated in detail in the text.

1 the "three-dimensional" one, provided that the line size is at

least four.

An HTS has the property that each plane is the nine point affine plane of order 3, while not being isomorphic to the point line incidence structure of some affine geometry over GF(3) •

Marshall Hall, Jr. constructed the first such system in [26] in i960. It has eighty-one points and is the unique HTS of minimum size. I should remark that in the literature such systems have sometimes been called affine triple systems; but since they are really only locally affine and since this treatise is by far the most extensive study of them, I have taken the liberty of naming them in Professor Hall's honor. The class of HTS’s in the case of line size three together with the class of

Steiner Quadruple Systems in the case of line size two show that Buekenhout's result can not be improved.

Also, there is at present a great deal of interest in perfect matroid designs. Quite a number of things have been proved about such geometries. For example, many designs, some of them new in terms of their parameters, can be derived from them. However, the only known examples of non-trivial perfect matroid designs are the classical projective and affine geometries, the t - (v,k,l) designs, and now the HTS's.

In 1965 Hall and B. H. Bruck realized that an HTS can be coordinatized by (and, in fact, is equivalent to) a coinnutative

Moufang loop of exponent 3* Bruck had developed an extensive theory of loops which in 1973 was exploited to some extent by

Peyton Young in [ 67 ] , a paper whi ch provided the impetus for this dissertation. Bruch's theory provides a wonderful algebraic tool with which to work, and the main goal of this treatise is to study the structure of HIS's using the equivalent loops.

By-products of this investigation are the construction of many new HTS's (and hence, many new derived designs) and a complete catalogue of the "small" ones.

Very recently HTS's have appeared as extremal examples of interesting geometries called Fischer spaces. A Fischer space is defined in terms of its automorphisms, and it is certainly possible that HTS's will have interesting automorphism groups. In this work we make a very modest start by computing the cardinality and some of the structure of the full automorphism group of Hall's original eighty-one point system.

Chapter I provides all the necessary definitions as we give the history of the interest in HTS's as special mathematical structures. They have arisen in the study of surprisingly diverse topics: cubic forms, differential geometry, and vector spaces in addition to loops, nets, matroids, permutation groups, and designs. We discuss their role, at least briefly, in each of these areas and mention some recent efforts (thus far unsuccessful) to construct new classes of Steiner Triple

Systems which would have similar regularity properties. Most of Bruck's very detailed theory of Moufang loops

appears in [6 - Chapters VII and VIII]. Since we and other

corabinatorialists have found this to be difficult reading, we

give in Chapter II a summarial account of much of this material,

most of which is needed to obtain the results which apply to

the special class of Moufang loops which are equivalent to HTS's.

The culmination of Bruck *s work is the Bruck-Slaby Theorem

which states that a commutative Moufang loop on d generators

is (lower centrally) nilpotent of nilpotence class at most d - 1 .

An immediate corollary of this is that an HTS must have 3n

points for some n > 4 . Heuristically speaking, the larger a

loop's nilpotence class is, the farther it Is from being

associative and hence, the less regular its multiplicative

structure is. Next we mention the work of Peyton Young, particularly

his construction of an HTS for each possible size 3n f n > ^ .

We also discuss in this chapter the recent contributions of

Lucien Beneteau, one of which is the analogue of the Burnside

Basis Theorem for finitely generated commutative Moufang p-loops.

Lastly we mention the new results of J. D. H. Smith and Jean-

Pierre Malbos which answer important questions originally posed

by Bruck in [6] and again in [7 ] .

Nearly all previously known examples of commutative Moufang exponent 3 loops (henceforth abbreviated by c-#7 e x p -3 loops) are subloops of an infinite loop, B, which Bruck constructed in

[6 ] > Because of the method by which they are constructed, the only obvious property of these loops is their nilpotence class.

In Chapter III we analyze their structure more closely, intro­ ducing the techniques which are used throughout the remaining chapters. We also point out the existence of a second infinite family of subloops of ft .

In Chapter IV we completely determine the structure of c-/7? exp-3 loops of nilpotence class 2. (c-fy exp-3 loops of nilpotence class 1 are elementary abelian 3-groups and correspond to the affine geometries over GF(3) .) We use this knowledge to construct an infinite family of loops which are irreducible in a sense to be made precise later. Each Irreducible loop in this family can be used to construct an infinite family of loops, none of which has been constructed previously.

In Chapter V we use the construction techniques of Chapter

IV and some additional structural results to obtain a complete catalogue of all HTS's of cardinality at most 3^ .

In Chapter VI we study the structure of c-/7f exp-3 loops of nilpotence class 3* We examine closely the case when such a loop is generated by four elements and give an explicit construction for the free c-7f\ exp-3 loop on four generators. We also prove a new identity for c-?7J loops of nilpotence class 3 .

In Chapter VII we take a brief look at some of the designs which can be derived from HTS's, discuss the Interest that the automorphism groups of HTS's may hold, and mention some directions far future inquiry. CHAPTER I

PRELIMINARIES : HISTORY AND DEFINITIONS

§ 1. Conventions and Basics

We adopt the following notations and conventions which will be used throughout this treatise. We remark that here and throughout Chapter I we give a definition of an object only if it is discussed in some detail, if it will appear in a subsequent chapter, or if it might be unfamiliar to the reader.

We denote by GF( q) the finite field with q elements, by AG(m.q) them-dimensional affine geometry over GF(q) , and by PG(m.q) the m-dimensional projective geometry over

GF(q) . We write W^(m,q) for the set of i-dimensional sub­ spaces of GF(q)m , the m-dimensional vector space over GF(q) .

If X is a finite set we denote by JxJ. the cardinality of X, by P(X) the set of subsets of X, and by Pk (X) the set of k-element subsets of X . If X and Y are sets then

X X Y denotes the Cartesian product of X and Y , and we write

X\Y for [x 6 X : x ^ I) , We shall always use N to denote the set (1,2,3,...) of natural numbers. We shall denote by (^) the standard binomial coefficient n!/m!(n - m) I .

We shall often write 1 for the identity element of a group.

We shall try to be consistent in our use of notation for mappings.

6 7

For example, when p is an element of the domain of a mapping

o we shall write pa for the image of p under a . The

symbol □ will be used to signify the end of a proof.

DEFIMITION 1.1.1; For given t,v,k, A € N, an ordered

pair D = (X, ft) consisting of a finite set X together with

a set ft of subsets of X is said to be a t - f v . k ^ ) design

iff

(i) |X| = v

(ii) |B| = k for all B € ft (i.e. ft c Pk (X))

(ill) For each T € *\(x ) there exist exactly A elements

of ft which contain T .

We sometimes say that a t - (v,k,A) design is a t-design when

we need not refer to specific parameters v, k, and A . Elements

of X are called points or treatments and elements of ft are

called blocks. We also write Jft| = b^ and for Y € *\(x ) with

1 < i < t - 1 we denote by r^(Y) the number of blocks containing

Y . For a more complete discussion of t-designs see [ 17] .

DEFINITION 1.1.2; A (v,k,A)-balanced incomplete block design

(henceforth abbreviated by (v,k,\)-BIBD) is a 2-(v,k, X) design.

The following proposition gives the arithmetic necessary

conditions for the existence of t - (v,k,A) designs. Farts (i) and (ii) were first proved by E. H. Moore in 1896 (see [^]) for the case of BIBD's. Because it illustrates the combinatorial

technique of counting the elements of a set in two ways, the

proof is included here. Part (iii) was first proved by

Fisher [23] in 19^0, also for the case of BIBD's (see [28]

for a proof). The general result follows because by part (ii),

a t - (v,k,X) design with t > 2 is a (v,k,r2 )-BIBD. When

t is even there is a better result which we shall discuss

next.

PROPOSITION 1.1.3 (Moore and Fisher): If (X,B) is a

t - (v,k,x) design then

(i) b - <>/(£)

(ii) P1(y) = *(tli>/(tli) for each Y € pi(x) 1 S i£ t"

so, we may Just write r^

(iii) If t > 2 and v > k then b > v .

Proof; To prfve (i) let Z = {(T,B): T € P^(X), B € B, and

T c B) . We have |Z | = (£) • X and also |Z | =■ b • (^) j hence,

I p y b(^.) *» X(^) and the result follows. To prove (ii) let

Z±(Y) - {(T,B): T € Pt (X), B € B , T c B , and Y c T) . For each T € P^(X) with Y c T (there are (^~*) of these) there are exactly X blocks containing T . So |z±(y)| = UjlJ) •

On the other hand for each block B with Y c B (there are k — i x^CY) of these) there are exactly t-element subsets of B which contain Y. So |Z^(Y) j = r ^Y ) and the

result follows. Since r^(Y) depends only on v, k, t, and i

we may write r^ for r^(Y) *

In particular we write r for r^ and in the case of a

(v,k,A)-BIBD we have b = A.v(v - l)/k(k - 1) and r = \(v - l)/k - 1 . □

Now we mention the much more recent result (1975) of

Kay-Chaudhuri and Wilson [ 50] •

PROPOSITION 1.1. h (Ray-Chaudhuri and Wilson): If s € N

y then for a 2 s-(v,k, X.) design with v > k + s , ^ > ( ) .

Of course by taking s = 1 in Proposition 1.1.^ we obtain

Fisher's inequality for BIBD’s. A 2s-(v,k,\) design is said to be tight iff b = (^) and in these terms the tight BIBD’s are the classical symmetric block designs. There is a wealth of symnetric block designs, but tight 2 s-designs for s > 2 seem to be very rare (see [1 ]).

We now define a class of BIBD's which have great importance for our work.

DEFINITION 1.1*5: S m (X, B) is a Steiner Triple System

(henceforth abbreviated by STS) iff it is a (v,3,l)-BIBD. We call v the order of S * The blocks of an STS are called triples 10

and a 3-element subset, A , of X which is not a triple is

called a triangle. Note that for an STS b = v ( v - l )/6 and

r - v - 1/2 .

For a good discussion of BIBD's see the book of Eyser [5 2 ] or the book of Hall [28]. The latter also has a special treatment of STS’s.

DEFINITION 1.1.6; t - (v,k,\) designs D = (X, 6 ) and

T' *» (X',fi*) are isomorphic iff there is a bisection o: X -*■ X* such that for all B € fi, BCT € fl* (i-e., a is a bisection between the point sets which takes blocks to blocks). Such a a is said to be an isomorphism from D to D r . An isomorphism, a , from D to D Is said to be an automorphism of D . The group of all automorphisms of D Is called the full automorphism group of D and is denoted by Aut(D) . A subgroup of Aut(D) is called a group of automorphisms of D . For an excellent study of automorphism groups of designs and of other structures as well, see [3 ] . a is called an involution of D iff a is an 2 automorphism of D and a = 1 where 1 is the Identity element of Aut(D) . x € X is a fixed point of a € Aut(D) iff xa = x .

DEFINITION 1.1.7: An ordered triple 3 = (P,£,l) where

F and £ are disjoint sets and I £ P x £ is said to be an incidence structure; it is said to be finite iff both F and £ are finite sets. Elements of F are called points and elements 11

of X are called lines. If (p, L) € I we say that p and L

are incident; and if lines h and L' have a common incident

point we say that L and L' intersect. Very often it happens

that X is a set of subsets of P and that (p, L) € I iff p € L ; in this case we shall write g = (F.X. €) and we shall

write p € L when p and L are incident. It also often

happens that P and X are sets of subsets of some given set

and that (p, L) € I iff p £ L ; in this case we shall write

S = (P,X,£) and we shall write p £ L when p and L are incident, g = (P,X,l) and g* = (P*,X',I*) are said to be isomorphic iff there exist bijections a: P -*■ P' and t : X ■* X* such that (p, L) 6 I iff (pCT, LT) € I* . We call an incidence structure linear iff for each pair, * of points there is a unique line, denoted by P-jP^ * which is incident with both p-^ and p2 . For a linear incidence structure (P,X,l) , a subset F c p is called a flat iff given any two points p1 and p2 of F , all points incident with PjPg are also contained in F . We sometimes say that F is closed under lining. For any subset Q, c p the flat generated by Q is denoted by (Q) and defined by (Q) = O F * Of Course (Q> is the F = Q F is a flat smallest flat containing Q since the intersection of flats is a flat. The geometric dimension of F . denoted dim (F) , is defined by dim (F) = min Iq [ - l . For example, a point Is Q C p - F 10

and a 3-element subset, A , of X which is not a triple is

called a triangle. Note that for an STS b = v ( v - l )/6 and

r - v - 1/2 .

For a good discussion of BIBD's see the book of Ryser [5 2 ]

or the book of Hall [28], The latter also has a special treatment

of STS's.

DEFINITION 1.1.6; t-(v,k,\) designs D = (X,8) and

D 1 = (Xr,fi') are isomorphic iff there is a bisection a: X -+ X'

such that for all B £ fl, BCT € 8 ’ (i.e., a is a bisection

between the point sets which takes blocks to blocks). Such a a

is said to be am isomorphism from D to D' . An isomorphism,

(x , from D to D is said to be an automorphism of D . The

group of all automorphisms of D is called the fun automorphism group of D and is denoted by Aut(D) . A subgroup of Aut(D)

is called a group of automorphisms of D . For an excellent study of automorphism groups of designs and of other structures as well, see [3 ] • oc is called an involution of D iff a is an 2 automorphism of D and a = 1 where 1 is the identity element of Aut(D) . x 6 X is a fixed point of a € Aut(D) iff xa = x .

DEFINITION 1.1.7: An ordered triple S = (P,£, I) where

P and £ are disjoint sets and I = P x £ is said to be an incidence structure; it is said to be finite iff both P and £ are finite sets. Elements of P are called points and elements 12 a flat of dimension 0 and a line is a flat of dimension 1 * A flat of dimension 2 is called a plane. If dim (p) = 2 we often say that (P, £, I) is a plane.

DEFINITION 1.1.8: An incidence structure * = (P,£, G) is a projective plane iff

(i) jr is linear

(ii) Two distinct lines of it intersect in a unique point

of it

(iii) There exist four distinct points of it, no three of

which are on the same line of it .

An incidence structure it = (P, £, G) is an affine plane iff

(i) it is linear

(ii) Each line of it is incident with at least two points

of it

(iii) dim (P) = 2

(iv) There is an equivalence relation, ||, on £ (called

parallelism) such that if p G P and L G £ then there

exists a unique line L* such that p G L 1 and L* || L .

Notice that part (iv) of the above definition is the analogue of the Parallel Postulate of Euclidean geometry. The equivalence classes of the equlvalence relation || are called parallel classes.

We see that by part (iii) of its definition an affine plane is a plane in the geometric sense. 13

PROPOSITION 1.1.9: A projective plane, (P, £, €), is a plane

in the geometric sense; that is, dim(P) = 2 .

Proof: Let jr = (P, £, €) be a projective plane. Of course by part (iii) of the definition of a projective plane, dim (p) > 2 .

We shall in fact show that P is generated by any three non- collinear points. Consider P-^PgjP^ £ P where p^ and let F = ) * We claim that F = P and since

F £ P by definition, ve need only show that given any p £ P , p € F as well. If p = p^ then certainly p € F . If p / p^ we consider the line pp^ : pp^ ^ p-^p2 since p^ £ p^pg . Hence, pp^ intersects P-jIt* ■’■n a sinSle point q where q £ F since all points incident with P-^Pg are elements of F . But

H ^ pp^ means that p € qp^ which forces p € F since F is a flat. □

We shall have occasion to consider only finite projective planes and finite affine planes, and in this case the planes are designs as described in the following standard propositions which we do not prove here.

PROPOSITION 1.1.10: (P,£, 6) is a finite projective plane p iff for some n > 2 (P,£) is an (n + n +1,n + 1,1)-BIBD .

PROPOSITION 1.1.11: (P,£,€) is a finite affine plane iff 2 for some n > 3 (P,£) is an (n ,n,l)-BIBD . 14

In both cases n is said to be the order of the plane. Very

often an affine plane of order 2 is also defined by P =

and £ = P2 (P) even though this incidence structure has geometric

dimension 3 •

DEFINITION 1.1.12: The point-line incidence structure of

PG(m.q) is (W1 (m +1,q),W2 (m + l,q),£) and the point-line

incidence structure of AGfm.q) is ({Wo (m,q) + v : v 6 GF(q)m },

{W1 (m,q)+v : v € GF(q)m},=) .

For example, the point-line incidence structure of PG(2,2) is just a projective plane of order 2 which is a (7,3,1)-BIBD and the point-line incidence structure of AG(2,3) is just an affine plane of order 3 which is a (9,3*l)-BIBD . Both of these designs are STS's . In the future we shall just say PG(2,q) instead of the point-line incidence structure of PG(2,q) and shall say the same for AG(2,q) .

Although the following definition is not standard, it will suffice for our purposes.

DEFINITION 1.1.13: A finite projective (affine) plane rr is

Desarguesian iff for some q 7 ir is isomorphic to PG(2,q)

(AG(2,q)) .

We remark that PG(2,2) and AG(2,3) are the unique STS's on 7 and 9 points respectively. Hence, FG(2,2) is the unique 15

projective plane of order 2 and AG(2,3 ) is the unique affine

plane of order 3 •

For a comprehensive treatment of projective planes see [ 33].

§2. Hall’s Original Work

We begin our historical account by discussing an important

paper [26] written by Marshall Hall, Jr. in i960. We will need

the following definitions.

DEFINITION 1.2.1; If X is a finite set then S denotes the group of all permutations of X . A permutation group is an ordered triple (X,G,Y) where X is a finite set, G is a finite group, and Y: G -► S is a homomorphism. We often Y identify G with its image under Y , writing g for g , and just write (X,G) for the permutation group. G is said to act on X and that G has a permutation representation in X . If

Y is injective the representation is said to be faithful. The degree of (X,G) is defined to be |X | . For a given k £ N

(X,G) is said to be k-transitive iff for any pair fx^.x^.... and (x^,x* x^) of distinct ordered k-tuples of distinct elements of X there is some g € G such that x® = for

1 < i < k . We often use the terms transitive, doubly transitive. and triply transitive for 1-transitive, 2 -transitive, and

3-transitive respectively. A permutation group (X,G) of degree n is said to be an fn.k) Jordan group iff it is doubly but not triply transitive and there is a subgroup H of G and a

subset Xjj ^ X such that |X^ | = k > 3 * x*1 = x for all

x € Xjj and for all h € H, and (xXx^H) is transitive. A

subgroup H satisfying these conditions is said to be a

k-Jordan Subgroup; it is said to be Jordan-maximal iff h is

contained in no k'-Jordan subgroup for 3 < k' < k and

tg € G: for all x € X xs = x} = H . H

Hall begins by noting the result of Ostrom and Wagner [^8 ] which states that if a finite projective plane has a group of automorphisms which is doubly transitive on its points then the plane is Desarguesian; so in general if a BIBD has a group of automorphisms highly transitive on its points, this can be considered as a "Desarguesian" property of the design. He then proves a result which we paraphrase as follows.

THEOREM 1 .2.£ (Hal],); If (X,G) is an (n,k) Jordan group with k-Jordan subgroup H which is Jordan-maximal then D = (X, ft) ia^ltn (n,k, 1)-BIBD where B = ({x € X: xS = x for all g € K} :

K is a conjugate of H in G} . Furthermore, G is a group of automorphisms of D which is doubly transitive on X , there is a block such that all elements of Bj, are fixed points of every h 6 H, and (xXB^H) is transitive. Conversely, if D = (X,B) is a (n,k,l)-BIBD which has a group of automorphisms

G such that (X,G) is transitive and for some B € B, (X\B,Hg) IT

is transitive where hB = {g€G:xs =x for all x € B), then

(X,G) is a Jordan group with k-Jordan subgroup H .

For the remainder of the paper Hall focuses his attention on STS's. In this case the Jordan group (X,G) which would correspond to an STS, S = (X, (ft), would be precisely a group of automorphisms of S such that the action of G on the set of ordered triangles of S would be transitive. By an ordered triangle we mean an ordered triple (x^,Xg,x^) of elements of

X such that is a triangle of S . The main result of Hall's paper is the following theorem.

THEOREM 1.2.3 (Hall): If S = (X,B) is an STS having a group of automorphisms G which is transitive on ordered triangles then exactly one of the following holds:

(i) Every triangle of S generates a plane isomorphic to

PG(2,2); in this case (X, (3, €) is isomorphic to the

point-line incidence structure of PG(m,2) for some

m > 2 .

(ii) Every triangle of S generates a plane isomorphic

to AG(2,3); in this case, if we make the extra

assumption that G is transitive on ordered tetrahedra

(a tetrahedron is a set of four non-coplanar points

and an ordered tetrahedron is an ordered If-tuple

(Pj^P^PgjP^) such that (P1,P2>P3,P^) is a 18

tetrahedron) then (X, fi, £) is isomorphic to the point-

line incidence structure of AG(m,3) for some m > 2 .

As we shall discuss more fully in §5 of this chapter, the extra

assumption in part (ii) is not needed for the conclusion to be

valid.

In proving this theorem, Hall obtains two intermediate

combinatorial results which we now state. Because of the very interesting interplay between the design structure and the automorphisms of the STS's involved, we give here a slightly modified proof of the second of Hall's results.

THEOREM 1.2.k (Hall): If S = (X, fl) is an STS with the property that for all B € fi there is an involution of S whose only fixed points are the points of B , then every triangle of

S generates a plane which is either isomorphic to FG(2,2) or isomorphic to AG(2,3) *

Hall also mentions that the converse of Theorem 1.2.If is false by noting that for m > *1 any involution of the point- line incidence structure of PG(m,2) must fix at least seven points. However, the converse for the case of an STS in which each triangle generates an AG(2,3) has not yet been decided. We should also mention here that a theorem of Teirlinck [58] which will be stated and discussed in §5 of this chapter prohibits the 19

existence of an STS in which all the planes are either PG(2,2)

or AG(2,3) and planes of both types are present.

THEOREM 1.2.5 (Hall): If S = (X,B) is an STS then for each x € X there is an involution, a . of S which has x * x ' as its only fixed point iff every triangle of S generates a plane isomorphic to AG(2,3) .

Proof: For this proof we will denote a point x^ £ X by i and the triple [i, j,k) will often be listed as just i, j, k .

We will also denote by i » j the third point in the block containing i and j .

First assume that S has the set of involutions

(a : x € X} . Notice that if (j,k) is a transposition of a. x a. a 1 then [j,k, j «k} 1 = tk,j,(j ok) ) which shows that has j o k as a fixed point, and hence j « K = i . Conversely, if a. i = j ok then consider the transposition (j,j 1) of ol which ct. a. contains j : by the above remark j o j 1 = i and so k = j 1 ; that is, (j,k) is a transposition of . Hence, we have seen that i, j, k is a triple of S iff (j,k) is a transposition of a. . l Let [1,2,If) be a triangle of S . Points 3, 5* 6 are determined by 1.2 = 3 ; I * 1* = 5 , and 2 o V = 6 . Then a point 7 Is determined by 1 o 6 = 7 * So far we have triples

1,2,3; 1,^,5; 1,6,7; and 2,^,6. From the involution °S =» (1 )(23)(^5 H &7 )• * • obtain the new triple (2,1j.,6 ) = (3,5,7 ) 20

and from the involution ct^ = (2 )(13)(46),.. we obtain the new OU a triple (1,4,5) = (3,6,5 ) • Now 1 ° 3 = 2 £ 6, 2 . 3 = 1 / 6, «P a? 7 0 3 = 5 / 6, 4 o 6 = 2 / 3, and 5 / 5 so 5 must be a

new point 8 from which another new point 9 is determined by

1 0 8 = 9 . We now have the following eight triples: 1,2,3;

1,4,5; 1,6,7 ; 1,8,9 ; 2,4,6; 2,5,8; 3,5 ,7 ; and 3,6,8 •

From ck^ = (1) (23) (^5 ) (67 )(09)» • - we obtain new triples by a, 0f. (3,6,8} A = {2,7 ,9 } and 12,5,8} A = (3,4,9) ■ Lastly from the ak involution = (4)(15)(2 6 )(39) we obtain (1,2,3) = (5 ,6,93

ai and then {5 ,6,9 } = (4,7,8) . Now we have the twelve triples

1,2,3 1,4,5 1,6,7 1,8,9

4,7,8 2,7,9 2,5,8 2,4,6

5,6,9 3,6,8 3,4,9 3,5,7 which constitute all the lines of the plane generated by 1, 2, and

4 (i.e. (1,2,...,9} is a flat) and which are the lines of an

AG(2,3) .

Next, to prove the converse we assume that each triangle of

5 generates an AG(2,3) . Let 1 be a point of S and consider the permutation “l of X which fixes 1 and interchanges x and y where l,x,y is a triple of S . It will suffice to show «i that B € B for each B € B . If 1 € B then of course «1 B = B € B ; so we consider a triple B = {2,4,6} which determines 21 triples 1,2,3; 1,4,5; and 1,6,7 . The triangle (1,2,If} generates an AG(2,3) which contains these first seven points and two additional points 8 and 9 as well; meaning that 1 ,8,9 ai is also a triple. We need only show that (2,4,6) is a triple of S , Since AG(2,3) is an affine plane, the third line in the parallel class containing 1,8,9 and 2,4,6 must be 3,5 ,7 ; and °S since = (1)(23)(45)(67)(89)... , (2,4,6} x = (3,5,7) 6 B. □

Hall merely invokes Theorem 1.3.2 and Theorem 1.3*5, the latter being implicit in the book of Veblen and Young [60], to prove the second statement of part (i) in Theorem 1.2.3* However, to deal with the second statement of part (ii), he does the following. He constructs an STS of order 8l which we henceforth denote by , As we shall see, this design has become a very important example - a sort of "Cantor set of finite geometries."

THEOREM 1.2.6 (Hall): ^*^8 ^ propertie s:

(i) The geometric dimension of X is 3 - *81 (ii) Every plane of is isomorphic to AG(2,3) .

(iii) Other than the design obtained from the point-line

incidence structure of AG(3,3), is the only

STS with properties (i) and (ii) .

(iv) There are sets X* c X^ and B* s fi, such that "81 "81 (X*,B',€) is an STS which is isomorphic to the point- 22

line incidence structure of AG(3,3) . (Hence,

|X'| = 27 and |B' | =117.)

Hall uses properties (iii) and (iv) above to prove the second

statement of part (ii) of Theorem 1.2.3 .

Before mentioning how was first constructed, we state

the definition of the objects which are the subject of this

treatise.

DEFIHITION 1.2.7: An STS S = (X,B) is a Hall Triple

System (henceforth abbreviated by HTS) iff (X,B,€) is not

isomorphic to the point-line incidence structure of any

AG(m,3) and every plane of (X,fl, £) is isomorphic to AG(2,3) .

In order to construct Hall first showed that any group with a number of the properties of the group generated by the set [a : x 6 X} of involutions described in Theorem 1.2.5 can be used to construct an STS in which every plane is isomorphic to AG(2,3) . Namely, such a group must be generated by Q involutions .. .,ctr satisfying (aj_aj) = 1 for all

(i,j) € Pg({l,2,...,r)) . He then constructs, using generators and relations, a group, K , with these properties which yields

. The order of K is 2 • 3^ and has a faithful representation which is a group of automorphisms of which is transitive in its action on X., *81 23

In a more recent paper [ 29] Hall gives a construction,

which we now present briefly, of a smaller group K* which

also yields . Let K* be the group presented by

K ' =

tbt = b”^ , tct = c ^ ) *

The subgroup, (a,b,c) , generated by a, b, and c is the

Burnside group B(3>3) and Is a normal subgroup of index 2

in K' . It is known (see [25], for example) that B(3,3) 7 7 has order 3 (hence, K' has order 2*3) and that an

element g £ B(3,3) has a unique representation in the form e. e_ e e, er e,- e g = a d c ^(a,b) (a,c) 'P(b,c) (a,b,c) ‘ where ei € [0,1,2 }

for 1 < I < 7 and the commutators (x,y) and (x,y,z) are

defined as usual: (x,y) = x"*^y”^xy and (x,y,z 1 = C(x,y),z) .

As usual cG(g) t "the centralizer of g in G is defined by

Cg(g) = [h € G: h gh = g} and [G:H] denotes the index of the

subgroup H in G . We shall denote C ,(t) by C . Hall J\ demonstrates that C = (t, (a,b), (a,c), (b,c)) ; hence, |C| = 2 .3^

and [K’:C] = 3^=81. He fixes a system [h^: 1 < i < 81} of

coset representatives: i.e., he takes K*/C = (Ch^: 1 < i <81 } ,

He then faithfully represents K 1 as a permutation group on the

81 elements of K ’/C by ifr; K r -► 'where g^: Ch -*-Chg .

Each of the 8l conjugates of the involution t in K' is represented by an involution fixing only one coset of C , and distinct conjugates fix distinct cosets. For example, t^ fixes C . By defining = (h^ 1 < i < 8l] and fi^ = {{h^h^h^}

(ChjjCh^) is a transposition of the involution fixing he

obtains the HTS .

§3* Characterization of Affine Spaces

We begin with a discussion of the classical axioms for projective spaces given by Veblen and Young in [60] .

DEFINITION 1.3*1: An incidence structure V = (P,£,I) is a

•pro.iective space iff it satisfies the following axioms:

P-l: T is linear.

P-2: If p, p^, and Pg are three non-col 1i near points and

and are distinct points such that p£ is

incident with pp^ and p^, is incident with ppg ,

then the lines P-jP2 and PjP2 must have a common

incident point. (See Figure 1.)

P-3: Every line L € £ is incident with at least three

points.

We note that P-3 was not originally includedj it serves to outlaw degenerate configurations such as a single triangle. P-2 is often called the Pasch axiom and is often phrased, "If a line intersects two sides of a triangle then it also intersects the third side." 25

Fig. 1: Pasch axiom

There is the following classical theorem of projective geometry (which we state for the case of finite projective spaces); it seems to have been proved first in a special case by Hilbert and then in greater generality by Veblen.

THEOREM 1.3*2 (Hilbert-Veblen): If r = (P,£,l) is a finite projective space such that dim(p) = d > 3 then for some prime power q, r is isomorphic to the point-line incidence structure of FG(d,q) .

Henceforth, for a projective space V = (P,£,I) we identify

L e £ with {p € P: (p,L) € I) and write r = (P,£, e) - As we shall presently see, in the case that r is a plane the axioms P-l, P-2, and P-3 are equivalent to the defining 'conditions of a projective plane given in Definition 1.1.8.

Let n = (P,£, €) be a projective space which is a plane generated by the three non-collinear points p, p^, and p2 , let L = p ^ g , and let P*(p,L) *= £p) U (s € P: there exists 26 ^ 'K' L € JE such that L = ps and L intersects L) * (P'fPjL)

is the set of all points which lie on some line through p which

intersects L .)

LEMMA 1.3 ■ 3 (Veblen and Young): P’(p,L) = (tp^p^Pg} ) = p.

Proof: To show that P'(p,L) is a flat we must show that

given ^ P'(PjL), P^Pg G p,(p>L) * This is immediate if

P^Pg = L or if Pj_Pg contains p and some point of L . Other­

wise there are two cases.

Case 1: p| £ L and p^ L .

Let q be the point of L on pp^ , and consider a point

r € P^Pg ‘ since rp intersects the line P-JPg in r and

the line p^q - p^p in p , by P-2 rp must intersect pjq = L .

So r € P'(p,L) and since r was chosen arbitrarily on P^Pgf we have p£p^ £ F'(p,L) .

Case 2; P-j^Pg £ L .

There exist points € L wi'th £ Pp £ and € pp£ .

By P-2 since PjPg intersects the line pq^ in p£ and the line pqg in pg , p£p^ must intersect = L in a point p^ . Now by the previous case, pjp^ £ P’(p,L) and since

P-JP^ = PjPg we have p£p^ c P'(p,L) .

Therefore P'(p,L) is a flat and certainly any flat containing p, p^, and p2 must contain P* (p, L) ; hence

P'(P,L) = (tp^P^Pg) > - P . □ LEMMA 1.3.**-: If it = (P, X, f:) is a plane then « is a

projective space iff * is a projective plane.

Proof: First assume that n is a projective plane.

Axiom P-l is immediate. Since jt has four points p^, Pg, p^,

and p^ no three collinear we consider the six lines they

determine: p1p2 and meet in a new point an^

PgP^ meet in a new point and P^P^ an(* ^2^3 mee^ a new point . So all six of our lines have at least three points; and let L be any other line. There is some point among p^,pg,p^,p^ which does not lie on L ; we can say w.l.o.g. that p^ L and the lines P^P^j ^1^3* and ^l^L in^ersect L in three distinct points. Hence, rr satisfies p-3 . And since every pair of lines of 7t intersect, P-2 is immediate.

Next assume that it is a projective space. By P-l n is linear. Since dim(P) = 2 , there exists a point p € P and a line L € X such that p £ L . There are at least three points p^, Pg, and p^ on L, and on the line pp^ there is some other point p£ . {p,p£,pg,p^J is a set of four points no three of which are collinear.

So, we must show that any two lines of jt intersect. Since it is a plane we know that P is generated by three non-collinear points p, p^, and pg ; let L = P^P2 * L01™181 1*3*3 we know that P = P'Cp,L) ; and furthermore, we have seen in the proof of Lemma 1.3*3 that each line of * intersects L . Hence, we

w.l.o.g. will always stand for "without loss of generality." 28

consider lines L-^ and 1^ of n which are distinct from L .

Here again there are two cases.

Case 1: p £ .

Let q he a point of and consider the line pq; pq,

L^, and intersect L in points r, p£, and respectively.

L1 intersects the line L = p^r in p£ and the line rq = pq in p ; so by P-2, L^ intersects p^q = .

Case 2: p ^ and p ^ L-, .

Let p£ and p^ denote the intersection of L with L^ and respectively. By P-3 there is a point p^ on L distinct from p£ and p£ . From Case 1 above, and both intersect pp^ , say in points q^ and q^ respectively. If q^ = q^ the result is established; otherwise we apply P-2; intersects the line L = in and the line p^q^ in q^ , so 1^ intersects P ^ ^ = ^

We now take note of the fact that the pasch axiom is a planar axiom: all points and lines mentioned lie in the plane

{{pjP^p^] ) . We can now deduce the following re suit,

THEOREM 1.3 -5 : If T = (P, £, €) is a linear incidence structure with dim(p) > 2 then r is a projective space iff every plane of r is a projective plane.

Proof: Assume first that r is a projective space and consider a plane n . Since axioms P-l, P-2, and P-3 all hold 29

for tt , we know by Lemma 1.3 A that n is a projective plane.

Conversely, assume that every plane of r is a projective plane.

P-3 follows immediately and to prove P-2 we consider the plane

it with points <{p,P1,P2 J ) * P^ and P2 are P°ints of it ,

so pjP2 is a line of jt which must intersect P1P2 • ^

We shall now discuss the efforts of Francis Buekenhout [12] in

1969 to obtain an analogue of Theorem 1*3*5 for affine spaces.

We first need to mention the axioms for an affine space. The

following set of axioms was used by Usa Sasaki in his important paper [53] published in 195^*

DEFINITION 1.3*6: An incidence structure r = (F,£,I) is an affine space iff

AS-1: T is linear-

AS-2: Using the definition that two distinct lines are

"parallel" iff they are disjoint and coplanar, if

p, q, and r are noncollinear points then there

exists a unique line incident with r and "parallel"

to pq .

AS-3: If f is a 3-flat of r and and are

distinct planes contained in F' such that

and jr2 have a common point, then and it2

intersect in a line L £ £ .

Sasaki also shows that in the presence of AS-i and AS-2, AS-3 is equivalent to the following axiom: 30

AS-3' : If L^ and 1^ are "parallel" and 1^ and ^

are "parallel" then and are "parallel."

Of course if one also defines any line to be "parallel" to

itself then "parallel" defines an equivalence relation on £ .

We paraphrase one of Sasaki's main results in the following

theorem.

THEOREM 1.3>7 (Sasaki): If V = (P,£,I) is an affine

space then there is an incidence structure r = (P,£, I) and

injections X -*■ £ and ^2: I -► I such that P c llfp______(p,L) € I iff (p,L ) = (p,L) , and r is a projective

space. (An affine space can be embedded in a projective space.)

From this one obtains a corollary which is the affine

analogue of Theorem 1.3.2 .

COROLLARY 1.3.8 (Sasaki): If r = (P,£,l) is a finite

affine space such that dim(p) = d > 3 then for some prime power q, T is isomorphic to the point-line incidence structure

of AG(d,q) .

Now Buekenhout uses a slightly different axiomatization of affine spaces due to Lenz [ 37l• (Here we again identify

L 6 £ with (p € P: (p, L) 6 I) and write p 6 L for

(P,L) € I - ) 31

DEFINITION 1*3.9: An incidence structure T = (P,£,l) is an affine space iff

AL-1: T is linear.

AL-2 : There is an equivalence relation on £, denoted

by |}, such that for each p € P and each L € £

there is a unique line L r such that p € L' and

L' |1 L*

AL-3 : If P-,P2 |1 P3Pif and p € pxp3 then either p €

or Pj_P2 and PPj^ intersect.

AL-4: Every line contains at least two points.

AL-5: If every line contains exactly two points then given

points p, p1# and p2, if p2 € 1^, 1^ || pp^ Pl e 1^ ,

and Iv> || pp2 then and 1^ intersect.

AL-6 : There exist two disjoint lines and I*-, such

that "H"I^ •

AL-6 is equivalent in the presence of the other axioms to dim(P) > 2 and can be discarded to allow an affine plane to be an affine space. AL-U should probably have been included by Sasaki to formally outlaw degeneracies. Buekenhout comments that one can show from these axioms that for distinct lines and I^, , || L-> iff and Iv> are disjoint and coplanar; so the two sets of axioms are essentially equivalent. It is also true that every plane of an affine space is an affine plane.

Hence, Buekenhout asks if it could be true that a linear incidence 32 structure of dimension at least two is an affine space iff all its planes are affine planes. Although he does not mention the case of line size two, we shall discuss it here.

DEFINITION 1.j.10: A Steiner Quadruple System (henceforth abbreviated by SQS) is a 3-(v,4,l) design.

In the case of an SQS the arithmetic necessary conditions of Proposition 1.1.3 amount to v = 2 or 4 modulo 6 . In a famous paper [32] published in i960, Haim Hanani proved the following theorem.

THEOREM 1.3*11 (Hanani): An SQS of order v exists iff v = 2 or 4 modulo 6 .

Now in an SQ3 S = (X,fi) one can define lines to be all the two element subsets of X and planes to be the blocks

(quadruples) of f3 . Then (X,P2 (X),€) is a linear incidence structure of dimension at least two (as long as v > 2 ) and has the property that every plane is an affine plane of order

2. But by Corollary 1.3.8 any finite affine space with line size two must have 2^ points. So we see that there are many

SQS's which are not affine spaces. A smallest such SQS has order v = 10 and we list its b = 120/4 = 30 blocks below. 33

1,2, 3,1+ 1,3,6,9 1,5,8,9 2,4,5,9 3,4,5,6 4,5,7,8

1,2,5,6 1,3,8,10 1,6,7,10 2,4,6,7 3,4,7,10 4,6,9,10

1,2,7,8 1,4 ,5,10 2,3,5,8 2, If, 8,10 3,4,8,9 5,6,7,9

1,2,9,10 1,4,6,8 2,3,6,10 2,5,7,10 3,5,9,10 5,6,8,10

1,3,5,7 1,4,7,9 2,3,7,9 2,6,8,9 3,6,7,8 7,8,9,10

So the analogue of Theorem 1.3-5 can not hold when the line size is two, but since this is somewhat of a degenerate situation

(since affine planes of order 2 are not geometric planes) this should be neither surprising nor bothersome.

Buekenhout then points out that Hall’s design shows that the desired analogue also does not hold when the line size is three; but then he proves the following remarkable theorem which mak.es HTS's all the more interesting.

THEOREM 1.3.12 (Buekenhout): If r = (F,JE, O is an incidence structure satisfying

(I) T is linear

(ii) Every plane of r is an affine plane

(iii) dim(F) > 2

(iv) Every line of V has at least four points then T is an affine space.

Buekenhout*s proof is a beautiful piece of work ninety per­ cent of which consists of proving the following lemma. LEMMA 1.3.13 (Buekenhout): Under the hypotheses of

Theorem 1.3*12, If jt is a plane of r, L' is a line of r

containing a unique point p of n , and V = (J Q, where \€A k

A = \ is a plane of r, X contains the line L* and x

contains some line of ir which contains p) and

Q, = (q € P: q is a point of X) then V is a flat. A

He completes the proof of the theorem by demonstrating the

transitivity of parallelism: to show that || L and L || 1^

imply L [J he applies the lemma with n being the plane

containing L and , L' being a line joining a point q

of L to a point q^ of I*, , and p being the point q

of L .

Thus, we do have an almost exact analogue of Theorem 1*3*5

with the HTS*s being the only truly geometric counterexamples

to the exact analogue.

§1*. Perfect Matroid Designs

In a pioneering paper [ 62] published in 1935 Hassler Whitney

first defined a matroid In an attempt to study the abstract properties of linear Independence. In the last twenty years there has been an ever increasing amount of combinatorial research in matroid theory. In many of our beginning remarks we shall follow the treatment given by Young and Edmonds in [ 65]. 35

DEFINITION 1.4.X : An ordered pair M = (E,J?) where ft ^ J a P(E) is called a matroid (an element I 6 J) is called an independent set) iff

1-1 : 0 € J.

1-2: If J £ I € J then J £ J.

1-3: For each A £ E all maximal independent subsets of

A have the same cardinality, which is denoted by

r(A) and called the rank of A , * r(E) is called the rank of M and is often denoted by r(M) .

A matroid M = (E,J) is said to be finite iff E is a finite set. (Since we shall only consider finite matroids we shall say matroid to mean finite matroid.) M is said to be simple iff P^(E) U P2 (E) ^ J. As is shown in [65] there is no real loss of generality in considering only simple matroids, so this will be our policy.

We already have an example of a matroid at hand. In a linear incidence structure S = (F,£, €) if we let = (I € P^fP)

I is not contained in any flat of dimension k - 2) and

lp! J = U JL then M = (P,J!) is a simple matroid in which k = 0 ^ g r(p) = dim(p) + 1 for each pap. (j^ is the set of triangles and is the set of tetrahedra.)

We continue with more standard terminology. DEFINITION 1.4.2: For a matroid M = (E,J?), D £ E is a

dependent set iff D £ J ; C ^ E is a circuit of M iff C

is a minimal dependent set; for A c E the closure of A

(in M) is denoted by cl (A) and defined by cl (A) =

A U (x ( E: there exists a circuit C of M such that

x £ C c A U (xjj ; A £ E is a flat iff cl (A) = A (flats are

often called closed sets); a k-flat is a flat A such tjiat

r(A) = k ; and H £ E is a hyperplane of M iff H is a

(r(E) - l)-flat. For A £. E an A-basis is a maximal independent

subset of A and an E-basis is called a basis of M . We denote

by a the set of all circuits of M , by M the set of all

hyperplanes of M, by 3^ the set of all k-flats of M, and

by 3 the set of all flats of M . The elements of 3^ are

often called points. Matroids M = (E,j) and M' = (E',J')

are isomorphic iff there exists a bisection a: E -+ E' such

that for all A E, Aa f J 1 iff A € J?. Note that in terms of flats a simple matroid Is one in which the O-flat is the empty

set and all the 1-flats are singletons.

Continuing with our prior example, for P £ P cl(p) = (P) , lines having at least three points are dependent sets, and a

set of three collinear points is a circuit. (The reader should be wary that what has been called a k-flat in the context of a linear incidence structure S is called a (k + l)-flat in the context of the associated matroid M .)

It is possible to axiomatize a matroid in many different equivalent ways: in terms of circuits, closed sets, hyperplanes, bases, and other things as well. We mention two such equivalent

sod.omatizations in the following propositions.

PROPOSITION 1.U.3: Let E be a finite set and consider

c3 = C iff

C-l: $ £ C-' and no element of C* is a proper subset

of another (a set with these properties is often

called a clutter).

C-2; If and C2 are distinct elements of Q} and

x 6 (1 Cg then there exists C € C* such that

C c (C^ U C2 ) \ tx} .

PROPOSITION 1 .tf : Let E be a finite set and consider

W s P(E) . There exists a matroid M *= (E,«0) with hyperplane set U = W* iff

H-l: E ^ W* and no element of H' is a proper subset of

another.

H-2: If and Hg are distinct elements of W* and

x € E then there exists H € H* such that

IxJ U (H^ Hit,) £ H .

For the proofs of these propositions see [ 6l] or [ 65] . We shall just mention how to recover the independent sets from C-' and Ji* . Given

]• Given W , let be the set of subsets of E which are 38

minimal with respect to being contained in no element H of

it* (fi will be the set of bases of the matroid) and let

*= (I c E : for some B € I c B) .

Two axiomatizations which are equivalent in the way that

1-1, 1-2, and 1-3 is equivalent to C-l and C-2 (or to H-l and

H-2) have been called (see [15]) cryptomorphic. For a complete

listing of all the cryptomorphic axiomatizations for matroids and their interrelations see [ I11 .

We now mention the matroids one can derive from a given matroid M = (E,j) = (E,Jt) .

DEFINITION l.U.5; M* , called the dual of M , is the matroid M* = (E,C*) where C* = (e \H: H G M) . (Note that because of the dual nature of the circuit axioms and the * . hyperplane axioms, it is immediate that M is a matroid*) * We see that the complement of a basis of M is a basis of M .

Complements of hyperplanes of M are called co-circuits of M •

For 0 < & < r(E) the ^-truncation of M is the matroid given by ^ *= (E,j/^) where |l [ < X) . When we (e) need not specify i we often just call M a truncation of M .

For E' £ E the reduction of M to E ' is the matroid given by M x E r *= (E’,Jlr) where =[l 6 -P: I £ E*} and the contraction of M to E 1 is the matroid given by M «E 1 = (E',J>c) where J>c = tl’ c E': there exists an (E\E’)-basis I such that

I ' U I e J) • For F1 € ff± and ^ £ 3^ with F1 = F* the 39

interval of M from F* to is the matroid given by

M(F^t ) = (M X F^) * (F^\F^) .

For much more comprehensive and deeper studies of matroids

see {15] and [ 61] .

The definition of an equicardinal matroid was first given

by Murty in [k'j] and served to link the study of matroids with

the study of designs.

DEFINITION 1.4.6: A matroid M = (E, M) is said to be an

equicardinal matroid iff all hyperplanes of M have the same

cardinality, denoted by k(M) . An equicardinal matroid in

which is a BIBD is said to be a matroid design,

abbreviated by MD .

This led naturally to the following definition first given

in [ 65] .

DEFINITION 1.^.7: A matroid M = (E,Ji) is a perfect

matroid design (henceforth abbreviated FMD) iff for all i

with O < i < r(M) all i-flats have the some cardinality,

denoted by ; that is, |F | = for aH F1 ( .

We shall see in an upcoming theorem that a FMD is always an MD ; and Young and Edmonds have constructed examples of

MD's which are not IMD's. Now one can always obtain a FMD by Uo

taking any truncation of a free matroid = (E,J^,) where

jlp = P(E) . Such a matroid is called a trivioid and a imd

which is not a trivioid is said to be non-trivial. As we

shall see, quite a bit of theory has been developed for PMD's;

and although the people who have worked in this area believe that there are many FMD's, the sad fact is that there are very few examples of them. In fact, the only known non-trivial

FMD's fall into one of the following three classes:

(i) The classical projective and affine geometries of

dimension at least three; FG(m,q) is a rank m + 1

PMD with aQ = 0 and cr = q1 - l/q -1 for

1 < i < m + 1 and AG(m,q) is a rank m + 1 FMD

with a = 0 and a. = q^ for 1 < i < m + l. o l — — Also, there are the truncations of these geometries.

(ii) The t - (v,k,l) designs with v > k: such a design

is a rank t + 1 PMD with = i for 0 < i < t - 1,

a, = k, and a. , = v . t 7 t + 1 (iii) The HTS’s; an HTS of order v is a rank *4- FMD with

a Q = ° , a ± = 1, a2 = 3, a « 9, and = v .

One of the best things about FMD's is that many designs, some quite interesting, can be derived from them (we shall give these constructions shortly). This is a major motivation for this dissertation, one of whose goals is to construct as many

HTS's as possible.

We take time here to mention two recent contributions to the study of FMD's. In [ Ray-Chaudhuri and Singhi obtain some arithmetic conditions and inequalities involving the

parameters |e|, k(M), and \ of a rank 4 MD, M = (E, W) .

They also prove the following two theorems. Because of its

elegance, the proof of the second is included here.

THEOREM 1.4.8 (Ray-Chaudhuri and Singhi): If M = (E,H) is a IMD of rank n > 4 then for each i and J with

2 < j < i < n , ac± - > (ag - l)(a± _ ± -a _ (*) .

Furthermore, equality holds In (*) for any i and j iff M is isomorphic to the n-truncation of PG(m,q) for some m > n - 1 or (3-^, J0 is a 3-design with - 2 .

THEOREM 1.4.9 (Ray-Chaudhuri and Singhi): If M = (E, h) is a rank 4 MD then M is a PMD.

Proof: Notice that since M has rank 4, (3^, W) is a

( |E |,k(M), \)-BIBD with \ > 1 . Let ^ be a 2-flat of M and let Z = {(x,H): x 6 E\p^, H € W, x € H and £ Hj .

On one hand |z| = \(k(M) - |l^ |) and on the other hand

|Z| = (|E| - |l^ | ) • 1 ■ Hence |l^ | = 1 is a number which is independent of the choice of . So, M is a PMD with ao = 0, = 1, a2 = I, = k(M), and a4 “ !EI * ^

In [ 18] Deza and Singhi give many nice results on the relation of PMD's to other topics in design theory. We give one example here. 42

DEFINITION 1.4.10: If k,v £ N and L is a set of non­ negative integers then an ordered pair (X, fl) is said to be an A(L,k,v)-system iff

(i) a = Pk(X) and |x| = v .

(ii) For distinct ^ ** lBi ^ B2 I ^ L *

Note that for a rank n FMD M = (E, W), an

A((“0'ai' • • - '“n - 2)<“n - l’“n )-system.

THEOREM 1.4.11 (Deza): There exists a polynomial VQ(X) such that if (X, B) is an A(L,k,v)-system satisfying |fi| = b V _ £ and v > vn(k) then b < Tf -- - (**) . Furthermore, & £ L 1 equality holds in (**)iff (X, ft) is a FMD (of course B is the hyperplane set of the matroid).

We now return to the work of Young and Edmonds. We mention that aii the material of [64] seems to appear again in either

[65] or [66] . The next theorem shows that PMD's have the same complete regularity found in the projective and affine geometries,

DEFINITION 1.4.12: In a rank n FMD for i, j, and k such that 0 5 ^ ^ £ n » ^ * and F^ £ 3^ with

F1 c J* we denote {F*5 € 3.: F1 c £ F11) by i.fF1,**) , The 0 J elements of are called the j-flats intermediate to F^ and . THEOREM 1.U.13 (Young and Edmonds); If M is a rank n

PMD -then for all i, j, and k with 0 < i < j ^or

all. F^ € 3^ , and for ai 1 F* € 3^ with F^ c F^, we have

|lj(F1, )| = tM(i,j,k) , a number which depends only on

i, jf and k, not on the particular choice of F* and F^ .

Furthermore, the function tM(i,j,k) , called the t-function

of M , satisfies

T-l: tM(i,i,k) = 1 for 0 < i < k < n .

T-2: tM(0,l,i) < t (0,l,i + l) for l

t (0,1,k) - t (0,1,1 ) T-3: t (1,1 + 1,k) = -H------for 0 < i

V 1>i'k)tM('e'3’k) T-^: t (i,j,k) - for 0

T-5 : ^(i, i + < ^(i, j,k) for 2 < i + 2

Note that t^(0,l, i) = .

One sees by the constancy of tM(2,n-l,n) that any FMD is an MD . Any positive integer valued function t*(i,j,k) which satisfies the conditions T-l - T-5 is said to be T-consistent and Is said to be realizeable iff t' = t.. for some IMD M . — 1— ™ ■■■■■■■ M There exist non-realizeable T-consistent functions; any prospective BIBD (viewed as a rank 3 FMD) whose parameters

satisfy the arithmetic conditions of Proposition 1*1.3 but whose existence is ruled out by the Bruck-Ryser-Chowla Theorem (see

[52]) provides such a function. kk

DEFINITION 1.4 .lU ; The d-seguence of "the rank n PMD

M is the ordered n-tuple (d ,d„ ...... d,) where d, = 1 and n' n - 1' 1 1 1 d± = t^O,1,1) - tM(0,l,i -i) = «i - a± _1 for 2

The d-sequence is an easier tool with which to work than

the t-function, and the two are related as follows.

THEOREM l.*f-.15 (Young and Edmonds): If t'(i,j,k) is a positive integer valued T-consistent function defined for integers i, and k where ° < i < j 5. ^ < n * anc* ^ d^ = t'(0,1,1 ) - t ’ (0,1, i - 1 ) for 2 < i < n and d^ = 1 then

J k / J J t 1 (i, J.K) = TT L d' / TT E d 1 i = i+l m = i / X = i + lm=i

for 0

DEFINITION 1 .h. 16: An ordered n-tuple (d *.d 1 ...... d„') .. - ■ ■ r ' n’ n - 1’ 1 1 ' of positive integers is said to be D-consistent iff d^ = 1 and the function t'(i,j,k) defined by the equation in the preceding theorem is T-consistent; it is said to be realizeable iff t'(i,j,k) is realizeable.

THEOREM 1.4.17 (Young and Edmonds): An ordered n-tuple

(d^,d^ is D-consistent iff

D-l: d^ = 1 and d^ > 1 for 2 < i < n . 5 k / J d D-2; TT Z d* / TT L d' is an integer for X = i + 1 m = X /X=i+lm=X

0

D-3: d.' is a divisor of dl for 2 < i < n and l-l i — —

d: d: 1 * ■ > V, 1 for 2 < i < n - 1 . di _di-l " "

The first statement of Theorem 1.J+.8 is an immediate

corollary of this theorem as is the set of necessary conditions

for the existence of a PHD in terras of its d-sequence.

COROLLARY 1.U.17 (Young and Edmonds); The d-sequence

(d ,d d, ) of a PMD is D-consistent and hence satisfies v nJ n - 1* * 1 ' D-l, D-2, and D-3.

We note that if M is a PMD with d-sequence (dn»dn then the truncation M (X) is a PMD with d-sequence

( L di#dJg _ * * *,dl^ and irrterval M(F^,E^) is a FMD i = X / ^k 'Sc-l + 1 * with d-sequence (-=----, -r , ... , ■ ■ ) * dj+l j+1 j+l

The classical PMD's can be completely characterized in terms of their d-sequences.

THEOREM 1.4.18 (Young and Edmonds): For t > 2, M = (E,h) is a rank t + 1 FMD with d-sequence (v - k,k - t +1,1,1,.. .,1) iff (3^,#) is a t- (v,k,l) design; for m > 3 and q > 2 M is a rank m + 1 PMD with d-sequence (qm ,q m -1 “ , ...,q,l)

iff M is isomorphic to PG(m,q) ; and for m > 3 and q > 2

M is a rank m + 1 PMD with d-sequence (qm ~^(q - 1), m - 2 q “ (q - 1),...,q(q - 1),q - 1,1) iff M is isomorphic to

AG(m,q) .

We now proceed to list the designs which can be derived

from FMD's. We shall discuss the particular case of HTS's

in Chapter VII.

DEFINITION 1A.19: For a rank n FMD M = (E,JJ) = (E,C-)

we write c(M) = min jc) for the smallest cardinality of a

circuit of M and we write |3(M) = c(M) - 1 . P(M) is called

the independence number of M because every set with p(M)

or fewer elements is independent. We define J}, = {i £ ji: |l[ = J)

and C. = [C € dx |C | = j) . Let = 1 , (in + 1 = 0, and

j -1

for p(M) < J < n let ^ Also for

j -2 0£n + .E f'1 ) i =1 p(M) < o < n + 1 let X. = J - P(M)

THEOREM 1.U.2Q (Young and Edmonds): Assume M = (E,»P) = (E,C) is a rank n FMD. 47

1, Given i and k with 3 < i +3

2. If p(M) > 2 then for each J with p(M) < j < n

(E, 3^) is a 0(M)-(an,aj,t^(p(M),j,n)) design.

3* If P(M) > 2 then for each j with p(M) < j < n

(E,Jj ) is a pCM)-{and,Uj) design.

1*. For each j with p(M) < j < n + 1, (E,d.) is a 3 p(M)-(QfnJd, Xj) design.

Wehave now seen much to motivate an effort to find new

FMD's. Young was thus led to his investigation of HTS's [67].

We shall discuss his work in §5 of this chapter when we begin our discussion of loops, but we mention here his general remarks on trying to construct new PMD's. By Theorem

1 .4 .1 8 , a rank 3 FMD is precisely an (ot^,a2,1)-BIBD , of which there are many examples. The natural procedure is to take such a rank 3 FMD and try to build it up to rank 4 by adding in

3-flats. One attempts to do this by using the free-erection process described by Crapo in (. . Of course it is a stringent requirement that all the new 3-flats have the same cardinality.

The only kinds of 3-flats which have been thoroughly studied are 2 the projective planes and the affine planes. Since n + n + 1 is never a square, there can be no "mixture" of plane types. So k&

we know from Theorem 1.3*5 311(1 Theorem 1.3*12 that the only

rank 4 FMD's we can obtain from these kinds of planes are

truncations of projective and affine geometries except in the

case of affine planes with line size three. This is the case of the HTS's, which at present have the status of being the only known non-classical FMD's.

Now, if we want to construct a PMD of rank n > 5 it is conceivable that such a thing could have a d-sequence of the 2 form (...,s ,s,l) without being a full projective geometry

PG(m,q) for sane m > n - 1 . However, as Richard Wilson pointed out to Young, it follows from Proposition 1.1.10 and

Theorem 1.3*5 that (3^, €) is isomorphic to the point-line incidence structure of some FG(m,q) so that s = q is indeed m + 1 _ a prime power and = — -- j— for some integer m > n - 1 .

A way by which one could achieve this goal would be to find in some PG(m,q) a subset E of the set of rank y + 1 flats

(where y > U) which partitions the set of rank k flats of

PG(m,q) ; this would result in a FMD with d-sequence m i y i 2 ( E q , Eq,q,q#l). Such a set £ would be a kind i =y + 1 i =3 of "generalized spread." Freeman and others have begun to investigate such things (so far most of the work has been done on sets of higher rank flats which partition the points of

FG(m,q)), and I believe it would be of great interest to find some. By examining the necessary conditions on the d-sequence 1+9 one sees that the "smallest" thing to look for is a set of rank

5 flats in PG(10,2) which partitions the rank h flats.

Lastly, we make the first original contribution of this treatise by pointing out an analogue of Wilson's observation.

PROPOSITION 1.4.21; If M is a PMD of rank n > 1+ with d-sequence (..s(s - 1 ),s - 1 ,1 ) for s > 3 then s is a prime power and a = sm for some integer m .

Proof: Every rank 3 flat of m is an affine plane of order s by Proposition 1.1.11 . Hence, if s > 4 then by

Buekenhout's theorem (Theorem 1.3-12), (3^, &,,€) is isomorphic to the point-line incidence structure of some AG(m,q) in which case s « q is a prime power and = sm . If s = 3 then (3 , 3^6) is isomorphic to (X, ft, €) where (X, ft) is an HTS which satisfies |x| = 3m (a deep result whose proof will be given in Chapter II); hence, |3jjJ = = 3m * □

In the case of s = 2 not covered by this theorem, it should be of interest to discover if SQg's can be erected to

IMD's of rank n > 1+ .

§9 . Coordinatization by Loops

The definitions given in this section are crucial to the remainder of this treatise. DEFINITION 1.5.1: An ordered pair (G, *>) where G is

a set and ° is a well defined binary operation on G is a

quasigroup iff for all elements a and b of G there is a

unique x £ G such that a ° x = b and there is a unique

y € G such that y <> a = b . When G is finite, |G| is

called the order of (G,«) . The quasigroup (G,«) is said

to be commutative iff a » b = b <> a for all a, b € G , totally

symmetric iff given that a ° b = c it must be the case that

b ° a = c, a°c=b, c<*a=b, b» c = a, and c»b = a ,

idempotent iff a o a = a for all a £ G , and distributive

iff a. (b»c) = (a ° b) » (a • c) for all a,b, c £ G . Note

that total symmetry implies commutativity.

DEFINITION A quasigroup (G, o ) is a loop iff there exists an element e € G such that eoa=a=aoe for all a £ G . Such an element e is called an identity element of (G,« ) and is unique since if e ' is also an identity element we have e * = e « e' = e . The identity element of a loop will always be denoted by e or by e„ ~ Cj if need be. We also shall write G for (G,o) when there can be no ambiguity. A mapping a: G H is a homomorphism from the loop (G. ») to the loop (H,Q iff for all a,b £ G

(a *b)a = aCTo ba . GCT is called a homomorphic image of

(G, *) . When a is a bijection, o is called an isomorphism and the loops (G, •) and (H,») are said to be isomorphic.

A homomorphism from (G, •) to (G, •) is an endomorphism of 51

(G, •) and an isomorphism from (G, •) to (G, •) is an

automorphism of (G, *) . Aut(G) and End(G) denote the group

of all automorphisms of (G, *) and the group of all endomorphisms

of (G, •) respectively. Since a loop (G, °) is a quasigroup,

for each a € G there are unique elements arJa£ € G such

that a o a^ = e and a^ ° a = e ; is called the right

inverse of a and a is called the left inverse of a . ------_JI ------Since in general a loop is not associative, it often happens

that a £ a . However, if a = a „ then we write a =a =a“^ t r z ’ t I r I --- which is called the inverse of a . This will be the case in

any commutative loop since then a„<>a = e= aoa =a ° a I T r which implies that a = a . A commutative loop (G, <>) is x r said to have the weak inverse property (abbreviated by w.i.p.)

iff for all a, b € G b®(a»b)^=a^". An arbitrary loop

(G, ») is said to have the left inverse property (l.i.p.) iff for each a £ G there is an element a* £ G such that for & all b f G, a', o (a o b) = b ; (G,°) is said to have the right X inverse property (r.i.p.) iff for each a £ G there is an element a^ £ G such that for all b € G, (b«a)»a^ = b; and

(G, ®) is said to have the inverse property (i.p.) iff it has both the l.i.p. and the r.i.p. We see by taking b = e in the above requirements that if a' (a1) exists then a' - a & r H & (a1 = a ) and that in an i.p. loop each element a has an r r inverse, a-1, satisfying a-1 <= (a«b)=b = (boa)o a"^ for all b in the loop. We have seen that a loop fails to satisfy only 52

associativity among the group axioms, and since a loop (G,°)

is a quasigroup we can define the associator, (x,y,z) , of

elements x,y,z € G to be the unique element of G satisfying

[x » (y o z )] o (x,y,z ) = (x » y) ° z . If (x,y,z) = e then

xo(yoz) - (x»y)»z and x, y, and z are said to

associate. A subset H = G is a subloop of (G, °) iff

(H, °) is a loop. The intersection of subloops is a subloop

and for A ^ G , the subloop generated by A is denoted by

(A) and defined by

To find interesting examples of quasigroups and loops we need only consider STS's. As is mentioned by Bruck in [10] and discussed by Evans in [ 20] there are two very natural ways of obtaining a loop from an STS (such loops have been called Steiner loops). We shall follow the treatment of Young [6 7 ] Quite closely. First we state a proposition whose proof

is immediate»

PROPOSITION 1.5-3 (Bruck): If S = (X,B) is an STS and

the operation o_ is defined on X by x « x = x and x ° y = z

where (x,y,z} 6 fi then (X, °) is an idempotent, totally

symmetric quasigroup. Conversely, if (X,») is an idempotent,

totally symmetric quasigroup of order at least three then (X,fi)

is an STS where fi = {(x,y,z): x / y and x ° y = z) . We call

• the quasigroup operation of s and denote (X,») by G(S) , which is called the quasigroup of S .

PROPOSITION 1.5A (Bruck): If S = (X,B) is an STS with quasigroup operation 0 , e € X , and the operation • is defined on X by x • y = (e = x) » (e ■> y) then (X,• ) is © © a commutative exponent 3 loop with the w.i.p. We denote

(X, ■_) by G_(S) and call it the loop of S based at e . © © Conversely, if G = (X, •) is a commutative w.i.p. loop of exponent 3 and order at least three with identity e then

S(G) = (X, fl(G)) is an STS where fi(G) = (tx~\y*"1,xy} : x and y are distinct elements of X) . Furthermore Ge(S(G)) = G .

(Note: the context will nearly always allow us to abbreviate

• by . and we shall even use Juxtaposition rather than • , so xy will mean x ■ y .)

Proof: First we consider (X, •). yx=(e«y)*(e®x)«:

(e o x) o (e * y) = xy since (X, <>) is totally symmetric; hence 5^

(X, •) is commutative, ex = (e ° e) » (e » x) « e ° (e » x) *= x

since (X,«) is idempotent and {e,x,e*x) 6 6 . So, e is

indeed the identity element of (X, *) . Since (X,°) is a

quasigroup we know that given x, z € X there is a unique

y 6 X such that (e « x) ° y = z and a unique y € x such that

e o y = y ; hence y is the unique element of X satisfying xy = (eox)o(e«>y) = z and since (X, •) is commutative, it

is a quasigroup and thus a loop. Now x • (e « x) *= (e « x) °

[e° (e»x)] = (eox)c.x = e so we can denote e ° x by x-^ .

Also x • x = (eox)°(eex) = e Q x a x_1 so (X, •) has exponent

3 • By observing Figure 2 below, we see that y(xy)”^ =

(y“1 )"1 (xy)"1 = y"1 o xy = x 1 , so (X, .) has the w.i.p.

Now, for the converse note first that the blocks described _1 2 are really triples because of exponent 3 : if we had xy = y = y then x = y which is not allowed and a similar conflict would “1 2 —1 —1 arise if xy = x = x . Since G is commutative, (x- )~ = x for al l x € X ; hence, y(xy) ^ = x-^- for all x,y e X implies that x-1 - (y-1 )_1 (xy)-1 and y"*1 = (x“1 )“"L(xy)_1 for all -1 —1 x,y € X . Thus, the triple of fl(G) containing x and y ip (x-1,y_1,xyj by definition, but it is also the unique triple containing y ^ and xy and also the unique triple containing x"1 and xy . Therefore S(G) is an STS , That G (S(G)) = G 6 is immediate. □

Figure 2: The Loop Based at e PROPOSITION 1.5*5 (Bruck): If S = (X,fl) ia an STS with

quaaigroup operation ° , e* / x, X'=XU{e'}, and the

operation * is defined on X* by e* *x = x*ef = x for

all x € X * , x * x = e ’ for all x 6 X* , and x * y = x o y for

distinct x,y 6 X then (X1,*) is a totally symmetric loop with

identity e* . Conversely, if (X*,*) is a totally symmetric

loop with identity element e' then (X’\(e'},fl ) is an STS

where ft* = t(x,y,z}: x and y are distinct elements of

X * \( e *} and x * y = z} .

Proof: First, by definition of * and the direct statement of Proposition 1.5*3 i"t is immediate that (X1,*) is a totally

symmetric loop with identity e' . The converse is also immediate by the converse statement of Proposition 1 .5 .3 * □

We denote (X1,*) by G (S) and call it the augmented loop of S . It is the loop C*e(S) which has been used almost exclusively to study the properties of the STS S . We can mention one result which involves the loop G (SJ . The proof is short and straightforward (see [39])*

PROPOSITION 1.5.6 (Macdonald): Assume that S = (X, ft) is an STS; then every plane of S is isomorphic to AG(2,3) iff in the loop G (S) for all distinct elements x, y, and z of

X such that x * y = z we have (x,y,z) = y . 56

We now give the definition of a Moufang loop which will

he of prime importance throughout this work. We shall always

write the loop product x • y as unless we need to do

otherwise.

DEFINITION 1.5.7: A loop (G, •) is a Moufang loop iff

for all x,y,z 6 G, [x(yz)]x = (xy)(zx) (771 ) . In the case of

a commutative loop this requirement can he written as

x[x(yz)] = (xy) (xz) (77\) . Equation (77f) (and equation (77l)

in the commutative case) is called the Moufang identity. We

shall, refer to equation (771*) as the g?*-identity and to

equation (Jas the ^-identity. We shall henceforth abbreviate

Moufang loop by 77j loop and commutative Moufang loop by c-77j loop.

The ^-identity is obviously a weakened form of associativity, but it is natural to wonder where it originated. 77{ loops are named after Ruth Moufang who was the first to study the ^-identity.

Apparently this identity first appeared in her 1933 study [J45] of what have come to be called Moufang planes. For complete details on this topic see [ 25 ]; very briefly, a Moufang plane is a projective plane which is a translation plane for each of its lines. In a paper published in 1935 [^6] Moufang proved the first results on 771 loops including what is now called Moufang's

Theorem which says that any 77[ loop is diassociative. We shall see this theorem again in Chapter II. 57

Very shortly thereafter 57? loops arose in a different context:

G. Bol's 1937 study [4] of nets^ which because of its appealing

nature we discuss here. This account of Bol's work is taken

from the excellent survey article of Bruck [10]-

DEFINITION 1*5.8: For n,k £ N with n > 1 and k > 3 an incidence structure 7? = (P,£,I) is an (n,k) net iff

(i) £ is partitioned into k non-empty classes (called k parallel classes) (that is £ = 1^1 £ ^ . i = 1

(ii) If lines L and L* are in different classes then

L and L ’ intersect in a unique point.

(iii) For each p £ P and each i with 1 < i < k there exists a unique line of £^ which is incident with p .

(iv) There exists a line L £ £ which is incident with exactly n distinct points.

The (n,k) net 71 is said to have order n and degree k . Nets of order 1 are trivial and so we shall assume that n > 1 ,

The following proposition is easily proved (see [10]).

PROPOSITION 1.5*9: If 7? is an (n,k) net with n > 1 then all lines are incident with exactly n pointst l£j - n for all i lp| = ^ > |^| - nk* k < n + 1, and k = n + 1 iff 71 is an affine plane.

Of course one finds many examples of nets by taking £ to be k of the parallel classes of an affine plane of order n . One can obtain quasigroups and loops from (n,k) nets; we

shall show the procedure in the case of k = 3 . (n, 3 ) nets

will be called 3-nets when we need not specify the order. For

1 < i < 3 we shall, call elements of i-lines. In our

diagrams we draw 1 -lines straight across the page, 2-lines

slanting down from left to right, and 3-lines slanting up from

left to right. For a 3-net of order n let G be a set with

|g| a n: the elements of G will be the elements of the loop

we construct. We begin by arbitrarily labelling the lines of

each parallel class with the elements of G . We shall denote the i-line labelled with x by (i,x) . We define a quasigroup operation as follows: x »y = z where (3,z) is the unique

3-line through the intersection of the lines (l,x) and (2,y)

(see the left diagram of Figure 3).

s (3jx): permute labels

(l,x): permute Nlabels

Figure 3: The Loop of a 3-Net 59

We shall now make (G, *) into a loop by merely permuting the

labels on the lines. Fix an element e € G and consider the

line (l,e): we permute the labels on the 3-lines so that for

each x 6 G , the line (3,x) is incident with the intersection

of the lines (l,e) and (2 ,x) (see the middle diagram of

Figure 3)* This guarantees that e *x = x . Lastly we

consider the line (2 ,e) and permute the labels on the 1-lines

so that the line (l,x) is incident with the intersection of

the lines (2,e) and (3,x) (see the right diagram of Figure 3).

Notice that the label e of the 1-line (1,e) is not disturbed.

Thus, we have insured that x • e = x and we have a loop

G, (7?) * Conversely, given any quasigroup (G, •) we can 1, e construct a 3-net 7J(G) as follows: for 1 < i < 3 let

= ((i,x): x e G), P - C(x,y): x,y € G), and declare a point

(x,y) to be incident with the lines (l,x), (2,y), (3,x *y) .

Now given our 3-net we consider the following operations on 1-lines (see Figure ^).

s

- - *i+p h

Figure U: Reflection of 1-lines 6o

Given 1-lines and 1^ and a point p on 1^ , let q

and r be the intersection of with the 2 -line through p

and the 3-line through p respectively. Then let s be the

intersection of the 3-line through q and the 2 -line through

r and define 1^ to be the 1 -line through s . 1^

is called the p-reflection of I*-, in 1^ , Reflections for

2 -lines and 3-lines are defined analagously, and we have the

following interesting results.

DEFINITION 1.5 -10: The 3-net 7? = (P, €) is

said to satisfy the Moufang-Bol axiom for 1-lines (respectively

2-lines and 3-lines) iff for each pair of 1-lines (respectively

2-lines and 3-lines) and 1^ is independent of

p (so we may denote the reflection of in by + Iv,) .

A 3-net which satisfies the Moufang-Bol axioms for 1, 2, and

3-lines is called a Bol net.

THEOREM 1.5.11 (Bol): If a net satisfies any two of the

three Moufang-Bol axioms then it is a Bol net.

THEOREM 1.5*12 (Bruck): If G is a loop then G is an

% loop iff 77(G) is a Bol net.

Bruck gives a lucid outline of the proof of the latter theorem in [10], appealing at the final stage to a theorem proved in [6 ] which we shall mention in Chapter II. 61

Also appearing in Bol’s paper is the first example of a

c-771 loop which is not a group. This loop was constructed by

Hans Zassenhaus and has cardinality eighty-one.

We now return to our discussion of 771 loops. We have

seen that the associator (x,y,z) is defined analagously to

the commutator (x, y) in the theory of groups. (See Definition

I.5.2 and the concluding paragraph of §2 of this chapter.)

One might hope for a theory of associativity in 771 loops or

perhaps in c-77( loops which parallels, at least to some extent,

the theory of commutativity in groups. This is precisely the

contribution that R. H. Bruck has made and we shall discuss a

part of his theory in Chapter II. In order to facilitate the

upcoming discussion we give the following definitions now.

They will be crucial to our study of the structure of 771 loops.

DEFINITION 1.5.13: If (G, •) is a loop, x € G , and

K £ G we denote [xk: k € K] by xK and tkx; k € K} by

Kx , A subloop H of G is said to be normal in G and we

write H A G iff for all x,y € G xH = Hx , (Hx)y = H(xy) ,

and y(xH) = (yx)H . When H A G we may define the quotient

loop G/H by G/H - ({Hx; x € Gj, •) where (Hx) • (Hy) = H(xy) .

The nucleus of G is denoted by N(G) and defined by

N(G) =* (a 6 G: for all x,y € G (a,x,y) = (x,a,y) = (x,y,a) = e) and the center of G is denoted by Z(G) and defined by

Z(G) = (z € N(G): for all x £ G zx = xz} . Z(G) is a normal subloop of G and if G is commutative then N(G) « Z(G) . 62

We shall need the following result which is part of

Theorem 2.1.10 which will be proved in Chapter II. It is also

an immediate corollary of Moufang's Theorem.

PROPOSITION 1.5.14 fMoufang); If (G, •) is an ty loop

then for all x,y £ G x(xy) = (xx)y .

Having given this preliminary discussion of % loops, we

shall now see the role that they play in the coordinatization

of STS's.

PROPOSITION 1.5.15: If S = (X,B) is an STS with quasi­

group operation o then every plane of S is isomorphic to

AG(2,3) iff (X,°) is distributive.

Proof: Assume that every plane of S is isomorphic to

AG(2,3) . By Theorem 1.2.5 we have for each x € X an

involution cz^ of S and by the proof of Theorem 1.2.5 we

know that (u,v) is a transposition of ax iff (x,u,v) £ (3 .

Given any y and z we shall see that x» (y«z) = (x o y) 0 (x ° z) .

This is immediate if y * x, z = x, z = y, or z = x 0 y since

(X,°) is idempotent and totally symmetric by Proposition 1 .5 .3 .

Otherwise we have a = (x)(y,x « y)(z,x 0 z)(y o z,x - (y ° z )) ... ax and (x»y,XoZ,(x(y) . (xoz)J ® ty,z,y <> z) j so x« (y«z) = a (y « z) = (x o y) o (x o z) . Second, assume that (X, 0) is distributive and consider the permutation which fixes x and interchanges u and v where {x,u,v} € (3. We show that 63

ax is an automorphism of S . Any triple of the form (x,u,v) ax satisfies (x,u, v) = [x,v,u) € B, so we have only to consider

a triple (y,z,yoz) which does not contain x . By distributivity °Sc ty ,z,y°z} b (x«y,x.z,x« (y«z)] = {x <> y,x o z, (x « y) a (x ° z)} €

B. □

In 1965 Bruck pointed out to Hall (see [2 7 ]) the direct

statement of the following theorem which has been stated in

its entirety by Young in [67] . We give here an improved version of Young's proof.

THEOREM I.5 .I6 (Bruck and Young): Assume that S = (X,B) is an STS. If every plane of S is isomorphic to AG(2,3) then for each e € X the loop Gg(S) is a c-77l loop of exponent 3 and furthermore, ai1 these loops are isomorphic.

Conversely, if for some e € X Ge(S) is an ^ loop then every plane of S isomorphic to AG(2,3) •

Proof: First assume that every plane of S is isomorphic to AG(2,3) ; as always we denote the quasigroup operation of S by o . Given e 6 X we consider the loop Gg(S): by

Proposition 1.5*^, G (S) is commutative and has exponent 3* By —1 Theorem 1.2.5 we have an involution of S, = (e)(u,u )(v,v )

(uv, (uv)"1 ). .. . Since {u"\v-\uv) € B we have

{u"1 ,v"1,uv}ae = (u,v,(uv)"1) 6 B and since by definition tu, v,u"’1v~1) 6 8 we see that (uv) 1 *= u~1v_i for all u,v € X .

We now verify the identity for commutative loops, (xy)(xz) = . 64

Cx“1y"1 )"*1 (x"1z“1 )“1 = (x ° y)_1(x o z )"1 = (x o y) • (x » z) =

x# (y “Z) , the last equality following from Proposition I.5 .I5 ,

Now x» (y*z) = x » (y^z"1) = x« (e«yz) = (x*e)o(x«.yz) = x"1 * (x»yz) again using distributivity. Finally x"1 • (x»yz) = x-1 ° (x^fyz)"1 ) = x[x-1 (yz)-1]-1 = x[x(yz)] . We remark that

since Ge(S) is an 771 loop, for all x,u,v € X, (xu)(xv) = x[x(uv)] - x"1 >* [x'^fuv)**1] = X _1 O [x ° (uv)] = (x"1 o x) O

[x-1° (uv)] - e«> [xfu^v"1 )] = e^tx^fuv)] = x-1 (uv) = (xx)(uv) -

(This certainly follows immediately from Proposition 1.5.14 , but I wanted to point out that it could be derived from the 771- identity and the distributivity of (X,«) .) Next, to prove that aii the loops are isomorphic we consider w 6 X\[eJ and show that G6 (S) -+ G,,(S) W is an isomorphism. We denote the -1 x ►> wx product of GW (S) by * and the inverse, w°x, of x in

G (S) by x . Note that by definition x *y = (w»x)« (w«y), Jr —1 i|f Jf —1 We see first that eT = we = w . If xT = y then wx = wy-1 which forces x"^ = y-^ and then x = y since Ge(S) is a quasigroup; hence is injective. Lastly

j* x _2. - l -l - 1 x ? *y T = (w o wx )» (wowy )s=w-> (wx ° wy ) by di stributivity

= wo [ (w-*^x)(w ^y)] = w o [ (w"^)’’^{xy)] by the remark stated above

r= w o [ w(xy) ] = w-1[w“1 (x"1y-1)] = (w"1 )"1 (x“1y"1 ) using the remark again

■ w(x-1y'1 ) = w(xy)-1 « (xy)* . 65

Hence ijr is an isomorphism.

For the converse, assume that G (S) is an ^ loop. By © Proposition 1.5*15 it suffices to prove that the quasigroup

operation of S is distributive. Given w € X we know

because of Proposition I.5 .IU that we can argue just as above

to show that Gw (S) is isomorphic to Gg(S) . Therefore we

have x *y = x *y and since x *y = x o y by definition of *,

we see that x *y = x 0 y which means that w o (x o y) = x *y =

(w 0 x) o (w ° y) again using the definition of * . Thus,

(X,e) is distributive. □

In our definition of HTS's we do not allow an affine geometry

to be an HTS; this amounts to not allowing the corresponding 171

loop to be a group. We shall need the following theorem which

is a corollary of the Bruck-Slaby Theorem and will be mentioned

in Chapter II.

THEOREM 1.5*17 (Bruck): If G is a finitely generated c~l7l

loop then Z(G) contains at least one non-identity element of

G .

DEFINITION 1.5*18: If (X,G) is a permutation group and x € X then the stabilizer of x in G is the subgroup of G given by Gx = £g € G: xg = x) .

PROPOSITION 1.5 *19 (Bruck-Hall); If S = (X,B) is an STS with quasigroup operation » and coordinatizing loop Ge(S) then Aut(Ge(S)) = Aut(S)e , the stabilizer of e in the action

of Aut(S) on X .

Proof: Given a € Aut(Ge (S)) , a: X X is a bisection

and furthermore (x o y)CT = (x^y-1)0 = (x-1 )CT(y-1)CT = (xCT)-1 (y<7)"1

xc o yCT ; and necessarily eCT = e . Conversely, given

a € Aut(S)g we know that a: X -> X is a bijection and also

that (xy)CT = (x-1 ° y-1)CT = (e ° x)CT o (e ° y)CT = (e ° xCT) o (e« yCT) =

(xCT)_1 . (y'7)"1 = x V \ □

In [S7 ] Hall omits the proof of the converse of the following

theorem and I found Young's proof in [67] to be unconvincing, so

I have included an original proof.

THEOREM 1 .5.20 (Bruck and Hall): An STS S = (X, fi) in which

every plane is isomorphic to AG(2,3) is isomorphic to the point-

line incidence structure of AG(m,q) iff Gg(S) is isomorphic

to the elementary abelian 3 -group . (C^ is the cyclic group

of order 3 •)

Proof: If S is isomorphic to the point-line incidence

structure of AG(m,q) then (X,Aut(S)) is doubly transitive

(see [17]) which means by Proposition 1 .5 *19 that Aut(Ge(S)) is transitive. By Theorem 1.5*17 there is an element z €Z(Ge(S))\(e}

Now recall the definition of the associator and note that if

[xCT(yCTza)](x,y,z)a = [ (xay°)za] which forces (x,y,z)CT = (xCT,y°,za) . Hence, z 6 Z(Ge (S)) iff zCT £ Z(Ge(S))

and since Aut(G(S)) is transitive we must have Z(G (S)) «= © 0 Ge(S) . Thus Ge(s) is a group and since it is commutative, and

has exponent 3, it is isomorphic to . Now for the converse,

assume that Ge(S) is a group. |x | = 3m and if m = 2 then

the result is obtained immediately. Hence, we assume that the

geometric dimension of X is at least three and we shall show

that the axioms of Sasaki (see Definition 1.3*6) are satisfied

by (X, (3, 6) i then by Corollary 1 .3*6 the result will follow.

Since each plane of S is an AG(2,3) the axioms AS-1 and

AS-2 are satisfied. We shall show that AS-3' is also satisfied.

That is, if triples = (x,w,v) and Bg = Ce,y,y-1} are

disjoint and coplanar in (X, 13,6) and if Bg and = {z,u,t}

are also disjoint and coplanar and also ^ B^ then B^ and

B^ are disjoint and coplanar. Let jr^ be the plane (B^ U Bg)

and let ng be the plane

not disjoint then would intersect itg in another point r

in addition to Eg which would force = Jtg ; but then we

would have two parallels to Bg through r which is impossible

because ir^ is an affine plane. Hence B^ and B^ are disjoint.

Now we may assume w.l.o.g. that {x,y,w~^} , (x,y-^,v’^) ,

tz,y,u”1) , and {z,y"’^,t"^J are triples of B . In terms of the group Ge(S) this says that x”^y"^ - w-^ , x”^y = v-^ , z o w by associativity. Hence t ° v = u<>x:=z0w and so

and are both triples of the plane generated by the triple

B^ and the point t <> v . □

COROLLARY 1.5.21: An STS S is an HTS iff G (S) is a

c-57| exponent 3 loop which is not a group.

It is historically amazing and interesting that Zassenhaus'

original construction of a c-V7{ loop which is not a group,

conceived in 1937t is precisely the loop that corresponds to

Hall*s HTS which was constructed more than twenty years

later. We shall mention this again in Chapter IV when we examine

Zassenhaus* loop.

We now discuss the impact of the Bruck correspondence of

Theorem 1 .5 .16 on the work of Hall discussed in §2 of this

chapter. By arguing exactly as in the proof of the direct part

of Theorem 1.5*20, Hall sharpens Theorem 1.2.3 to the following.

THEOREM 1.5.22 (Hall and Bruck): If X = (X,fi) is an STS

having a group of automorphisms which is transitive on ordered

triangles then (X,B, €) is isomorphic to the point-line incidence

structure of PG(m,2) for some m > 2 or to the point-line incidence structure of AG(m,3) for some m > 2 .

At the same time (1965 - see [27]) Hall conjectured that the conclusion of Theorem I.5.22 would still be valid if one only 69

assumed that S = (X, fi) had a group of automorphisms doubly

transitive on X . As far as I know this conjecture is still

unsettled- There is a related work of Teirlinck [98] which was

published in 1975 and improves the result of Theorem 1.2 A .

THEOREM 1.5.23 (Teirlinck): If S « (P,£, €) is a linear

incidence structure in which every plane is either affine or projective then either all planes of S are affine or all planes of S are projective.

An STS S = (X, fl) is called a Steiner space iff dim(X) > 3 in viewing (X, R, €) as a linear incidence structure. Teirlinck uses Hall's Theorem 1.2.6 in conjunction with Theorem 1 .5.23 to classify the Steiner spaces of order at most 2 7 .

Young did more with loops in [67] but we postpone discussion of this until we have seen more of Bruck's theory.

§ 6 . HTS * s in New Contexts

We first discuss the interesting geometric properties of

viewed as a rank 4 PMD with aQ = 0, ^ = 1, = 3, = 9, and = 8l and of the PMD's corresponding to general HTS's.

These results were proved by Young in [6 7 ]. In this section of his paper Young points out that in 1971 in a lengthy study [5 7 ] of the operation of taking the arithmetic mean of points in a vector space, Jean-Pierre Soublin had encountered algebraic structures which are equivalent to c-7?? exponent 3 loops.

Soublin calls these structures medial spaces and constructed

a medial space equivalent to Zassenhausf eighty-one point non-

associative c-^f exponent 3 loop. Soublin and Young each

demonstrates in his own setting that is the unique HTS

of order 8l, a fact which is implicit in Hall’s original work.

DEFINITION 1.6.1: If S = (X, fi) is an HTS then B € fi is

a central line of S iff it satisfies transitivity of "parallelism

that is, for distinct £ S 3 "parallel" to B and B

"parallel" to B2 imply B.^ "parallel" to B2 . (Here "parallel"

means disjoint and coplanar.) A subset p c fi is a central

class iff ^ partitions X and is "parallel" to B2 for

all distinct € ? • A subsystem S ’ = (X',B') of S is

a box iff (X*,BT,6) is isomorphic to the point-line incidence

structure of AG(3*3) • We denote the set of boxes of S by )

Since B^ is "parallel" to B iff for same x £ X\B

B^ = Bx (a coset of B in Ge(S)), the lines of a central class

are central lines and every central line is in a unique central

class. The proof of the converse part of Theorem 1.5.20 actually

shows that in an HTS S = (X, fi) , B = (e,y,e » y) £ 13 is a central line of S iff y €Z(G (S)).

When the choice of e is immaterial we shall henceforth denote the loop Gg() by Gg^ . It is a fact (proved in 71

Chapter II) that JzCG^) j = 3 • If S = (X, 13) is any HTS and

a: G © (S) ■* G w (S) is the isomorphism given in the proof of Theorem 1.5.16 then since (x,y,z) = e for all x,y € X iff (x,y,zCT) = w

for all x,y 6 X, it follows that Z(G (S)) = Z(G (S))CT. We also W 6 see that (e,w,w ^]a = (w, e,w . Thus, the center of each of the

eighty-one loops of *81 is the unique central line containing the identity element of that loop, and a central line occurs as a center exactly three times; hence Jtg^ has a unique central class of twenty-seven central lines which we denote by Pq ^ .

THEOREM 1.6.2 (Young): Every box of = (Xg^, contains 2rj points of Xg1 ; = 39 \ given x^x^ 6 Xg^, if tx^,x2 ,x1o x^] f p^^ then x^ and x^ occur together in exactly 13 boxes and if {x^,Xg,x^ ° x^) $ pQ^ then x^ and Xg occur together in exactly 4 boxes.

In other language (see [49]) this theorem states that (Xg^,c7()Jg^)) is a (v,b,r,k,X^, \g)-partially balanced incomplete black design with v = 81, b = 39, r = 13, k = 27, X1 = 13, and v, = 4 .

This is proved by showing that with TT being the set of planes of *81 which contain some element of (%l'n'c> ls isomorphic to the point-line incidence structure of AG(3,3) .

DEFINITION 1.6.3: A tetrahedron T = [w,x,y,z) of the HTS

S *s (X, fi) is said to be singular if the points w, x, y, z generate a box of S and non-singular otherwise. The midpoints 72

of T are the points (v«x). (y0 z), (woy)o(x0 z), and

(w o z) o (x o y) .

THEOREM 1.6A (Young): Given a tetrahedron T of an

HTS S , if T is singular then the midpoints of T are

identical and if T is non-singular then the midpoints form

a central line of S .

Part of this theorem had already been proved by Mitschke

and Werner [43] when they deduced a characterization of affine

geometries among STS's as a corollary of a more general theorem of theirs in universal algebra.

In his review [34] of Young’s paper Kantor mentions some more recent work that has been done on 7% loops ([2 2 ], [24], and

[ 42]) in which other proofs of the fact that an HTS has order

3™ appear. However, in each of these the alternative to using the Bruck-Slaby Theorem is invoking the Feit-Thompson Theorem.

In his book [42] Manin especially points out the correspondence between the c -77{ loops and certain groups which are examples of Fischer groups and the resulting interplay between the loop theory and the group theory. We shall define Fischer groups shortly, but we mention that in this context the

Fischer groups that appear are precisely those which were originally studied by Hall: the groups generated by the involutions : y ** x ° y for x a loop element and o the corresponding quasigroup operation. Manin also points out that 73

ether Burnside groups B(3,r) where r > 4 can be used in

the manner of Hali (see the last paragraph of §2 of this

chapter) to construct non-associative c - ^ loops of exponent

3 •

We have already seen in Proposition 1*5.15 that distributive

quasigroups are intimately related to HTS*s. Properties of general distributive quasigroups were first investigated by

Fischer in 1964 (see [21]). His main theorem is the following.

THEOREM 1.6.5 (Fischer): If Q « (X,«) is a distributive quasigroup then the group G(Q) = (a : x £ X) is solvable where : y *-► x o y .

Now if a distributive quasigroup is idempotent and totally symmetric then by Propositions I.5.3 and I.5 .I5 it is the quasigroup operation of an STS in which all planes are isomorphic to AG(2,3) • This kind of quasigroup has arisen in two surprising areas: first in Manin’s study of rational points on a cubic surface and second in the work of Loos [38] in the field of differential geometry. In the latter these quasigroups are called kommutative Spiegelungsrflume (commutative reflection spaces). We have recently been informed of the dissertation

I believe that in the common areas of investigation with regard to both method and results, this treatise is a definite improvement to Klossek's work. In the part of her work that I have read thoroughly I have found at least one error of a non-trivial nature. of Sonja Klossek [35 ] which discusses these from a purely algebraic

point of view, often transferring Bruck's results on loops

to the quasigroups via ,the correspondence of Proposition 1 .5 .4 .

DEFINITION 1.6.6: If (G, .) is a group and A s G then

the ordered pair (A,G) is a Fischer group (we say that G

is a Fischer group when A need not be specified) iff

(i) For each x € A x is an involution.

(ii) G is generated by A .

(iii) If x £ A then for all y € G y”^xy € A .

(iv) For each pair of distinct elements x,y £ A the

element xy is of order 2 or of order 3 •

The three new sporadic simple groups Fi22, Fig^ and Fi^ discovered by Fischer are all of the form G/Z(G) where G is a Fischer group. Since the Mathieu groups Mpp, M^, and are groups of automorphisms of very interesting geometries

(see [3]), Buekenhout has begun the study of the geometries of Fischer groups. The following material is taken from [13]*

From a Fischer group F = (A,G) with identity element e one obtains a linear incidence structure £(F) = (A,£,£) by setting

X a t(x,y} : X / y and (xy) = e) U t(x,y,xyx] : (xy)"3 = e} .

For x € A the mapping a^: A ■+■ A is an automorphism of y h. xyx

£(F) which fixes x, fixes z if {x,z) € and Interchanges 75

z with xzx if (x,z,xzx) £ £ . So, Buekenhout makes the

following definition.

DEFINITION 1.6.7: A finite incidence structure g = (A, £, f) is a Fischer space iff

(i) g is linear.

(ii) If L € £ then either |l | = 2 or |L| = 3 -

(iii) For each x € A the mapping a : A -*■ A given by a a a x x x = x,y X = y if (x,y) € £ and y x = y' if (x,y,y'} € £ is an automorphism of g .

The Fischer group of g is given by F(g) = (A, (a^: x € A)) with the group operation being composition of mappings.

The proof of the following proposition is immediate.

PROPOSITION 1.6.8 (Buekenhout): If g is a Fischer space then F(g) is a Fischer group and £(F(g)) is isomorphic to g .

We see that the HTS's (and also the geometries AG(m,3)) are extremal Fischer spaces in the sense that ai 1 the lines contain three points. It seems likely that the Fischer groups of HTS's (we have noted in §2 that |F(Mq-|_) ( = 2 • 3^) and the full automorphism groups of HTS’s (we have also seen that jAut( ) | > 2 *310) will be interesting groups. We conclude our history of HTS's by mentioning recent

efforts (thus far fruitless) to construct rank 4- PMD's which

are similar to HTS's. My own effort has been this: since the

two smallest non-trivial STS's, PG(2,2) and AG(2,3) , are

planes of rank if PMD's, I have tried to use as a plane the next

smallest STS. There are two non-isomorphic STS’s of order 13

(see [28]), both of which are planes. One of them which I

shall call T = (X^,,^) is regular in that it arises from a

difference family in the additive group of integers modulo 1 3 :

X T = (0,1 ,2 ,...,1 2 }, f) = U where = { ( 0 + i, 1 + i, 4 + i)

0 < i < 12} and = {{o + i, 2 + i, 7 + i]: 0 < i < 12} . I

shall denote the other STS of order 13 by T' . I have found

the fni i automorphism groups of T and T' by a computer

search.

PROPOSITION 1.6.9: Aut(T') is S^ , the non-abelian group

of order 6 , and Aut(T) is the non-abelian group of order 39

generated by c: x •+ x + 1 and m: x 3x , the multiplication being performed in the ring of integers modulo 1 3 . (Of course,

3 is called a multiplier; see [30] and [36] for the theory of multipliers.)

Because T is more regular than T' I looked for an STS

In which all planes are isomorphic to T * (Perhaps It would be better to allow both T and T ' to appear as plane s.) The arithmetic necessary conditions for the order, v, of such an 77

STS as given in [ 18] are v e 3 , 1 3 , 133, 183, 2 7 3 , or 313

modulo 390* However, these same considerations for the case

of planes being AG(2 ,3 ) lead only to the requirement that

v = 3, 9 > 2 7 f 5 7 > 75j Si, 99, or 129 modulo 1^4 whereas we

know that the only real possibilities are v = 3m for some

m 6 N j hence, it is certainly possible that there are no STS's

of the type desired. Another pessimistic remark is that even

T has much less symmetry than AG(2 ,3 ) since |Aut(AG(2,3 )) | = ^32 .

Michel Dehon has looked for STS’s with even more regularity

(see [16 ] ) .

DEFINITION 1.6.10: An STS S = (X, 13) is said to be planar

iff all planes of S have the same cardinality, and a planar

STS is said to be a - regular iff for every plane TT of S and every triple B € (3 which does not intersect TT there exist exactly a planes TT-^, ..., of S such that for

1 < i < a , Tr± contains B and the intersection of TT and Tf^ is a triple of (3 •

THEOREM 1.6.11 (Dehon): An 0: - regular planar STS must be one of the following:

(1) The points and lines of PG(m,2) for m > 3: a = 0 and all planes are PG(2,2) .

(2 ) The points and lines of AG(3,3): ck = 3 and a n planes are AG(2,3) . (3) An STS of order 2(6m + 7) (3m2 + 3m + 1) + 1 where m > 1: a = 1 and each plane ie of order 6m + 7 (note that when m = 1 the planes have 13 points and v = 183).

(4) An STS of order 171: a = 2 and each plane is of order 15.

(5) An STS of order 1 8 3 : ct = 7 a»d each plane is of order 21.

(6 ) An STS of order 2 0 5 5 : a = b and each plane is of order 39*

This theorem is followed by the sad statement, "No example of type (3), (4), (5 ), or (6 ) is known."

Because of what we have seen in §U such an STS, and in fact any new rank U PMD, will be of great interest if it exists

I hope this chapter has shown that HTS*s are interesting objects and are worthy of study. CHAPTER II

AN ACCOUNT OF BRUCK'S THEORY OF MOUFANG LOOPS

§1. Moufang Loops

It was no accident that It was R. H. Bruck who first pointed

out to Marshall Hall the correspondence between HTS's and c-7%

exponent 3 loops (from now on we shall always abbreviate exponent

3 loop by exp-3 loop ahd the reader is warned that in some of

the literature, most notably (67], exp-3 loops have been

mistakenly called 3-loops - see Definition 1 .5 .2 for the

distinction); for it was Bruck who in the 19^0's and 1950's

had been the principal architect of the theory of loops and the theory of 7% loops. The first large piece of his work [5 ] entitled "Contributions to the Theory of Loops" was published in 19^6 and then in 1958 came his book [6 ] "A Survey of Binary

Systems" in which five of the eight chapters are devoted to loops; this was followed by two shorter papers [8 ] and [9 ], both published in i9 6 0 . These last two are not used in this treatise and we shall only comment briefly on [5 ]; the main purpose of this chapter is to summarize those results of Chapters VII and

VIII of [6 ] which have the greatest application to the combinatorial structures (HTS's, equivalent by Corollary 1.5.21 to non-associative c-771 exp-3 loops) with the hope that combinatorialists can use this

79 catalogue as an easy reference. It will "be worthwhile to recall

the definitions concerning loops which have already been given,

namely Definitions 1 .5 .2, I.5 .7 , and 1.5.13 . All propositions

and theorems in this chapter are due to Bruck unless otherwise

noted.

One of the main topics of [5 ] is the development of an

extremely general theory of nilpotence in loops. The special

case of central nilpotence will be touched on in §2 of this

chapter. Much of the material in [5 ] on Wi loops is treated

again in Chapter VII of [6 ], but a very important result that

does not reappear is the following. As was pointed out by

Beneteau in [2] it is only implicitly stated by Bruck.

THEOREM 2.1.1: If G is a finite c-SPj loop, O^G) = (x € G:

o(g) = 3m for some m > 0} , and = {x € G: o(g) and

3 are relatively prime) then 0^(G) is an abelian group, 0^(G)

is a c-fli 3 -loop and G *= ^3 ^ ) ® ^3 ^^ ^we write H = <£■ H-, to indicate that H is the direct product of

and Hg).

This theorem says that every finite c~77{ loop is the direct product

of an abelian group and a c-W{ 3 -loop; it is the first of several indications that the number 3 has a very distinguished role in the theory of c-2?j loops. The structure of finite abelian groups is completely understood by virtue of the Fundamental Theorem (see

[2 5 ], for example); so we shall have full knowledge of finite c-ty 81

loops when we have come to understand non-associative c-57?

3-loops. Furthermore, notice that those loops which are equivalent

to HTS's (the non-associative c-Wf exp-3 loops) form a significant

subset of the set of non-associative c-/7j 3-loops.

When we consider a class C of loops (for example, Q,

could be the class of all c-#f loops on d generators), by the

free element of C we mean the unique (up to isomorphism) loop

F € C having the property that all loops of C are homomorphic

images of . For example, we shall see that Gg^ is the free

c— exp-3 loop on three generators.

As I have mentioned before, [5 ] is very lengthy and very

detailed, I have just recently discovered that Bruck's Theorem

9A of Chapter II gives a construction of the free c-7% loop of

nilpotence class 2 on d generators (nilpotence class will be

defined in §2 of this chapter and will be a key tool from there

on) which implicitly contains a construction of the free c-TTj exp-3

loop of nilpotence class 2 on d generators for each d > 3 .

This is part of my own Theorem ^.2.5 which I have considered an

important contribution of this dissertation. Although I was unaware of Bruck's construction when I conceived and proved this theorem, I think it must be viewed as an extension of Bruck's result in that all the homomorphic images of the free c-ty exp-3 loop of nilpotence class 2 on d > 3 generators can be listed as a consequence of my theorem. And it is these loops that prove to be most valuable in the explicit construction of new HTS's. 82

We now proceed to our summary of Chapter VII of Bruck's

book. Many of the proofs of these results are both ingenious

and complicated, and only rarely do we include one.

Loops with the inverse property (i.p. loops) are a basic

class of loops which have the worthwhile property of being closed

under the operation of taking a homomorphic image. Having already

seen that % loops arise naturally in the study of STS's and

the study of nets, we shall now see that they also play a

distinguished role in the study of i.p. loops. From now on,

the loop (G, •) will be assumed to have the i.p. The composition

of a mapping a followed by a mapping t will always be written

ctt , i.e. we compose from left to right.

DEFINITION 2.1.2; For an i.p. loop (G,») we define the

following mappings from G to G : 1^: x x , R(x): y •+ yx,

L(x): y xy, J: x x-1, F(x): y ** (xy)x, T(x): y *-*- x-1 (yx),

R(x.y): z m- [ (zx)y] (xy)-1, and L(x,y) : z ►> (yx)-1[ y(xz) ] . The

multiplication group of G is denoted by Jlr(G) and is defined

to be the group generated by (L(x): x € G) U (R(x): x € G) .

3(G) will denote the inner mapping group of G and is the sub­

group of J^G) defined by 11(G) = .

Because G has the i.p., G) is in fact a group in which

L(x)-1 - L(x-1) and R(x)-1 = R(x-1) . We see immediately that

P(x) - L(x)R(x), T(x) = E(x)L(x)-1, R(x,y) = R(x)R(y)R(xy)“1,

L(x,y) = L(x)L(y)L(yx)-1, JJ = IQ, JL(x)J = R(x)-1, and 83

JR(x)J=L(x) . To see that the last two equations hold we

need only observe that z = xy iff x~^z = y iff = yz-^

iff y ^x"1 = z-1, meaning that for all x,y £ G (xy)"1 - y'^x-1 .

3"(G) is an extremely important tool throughout Bruck's work; in

the special case that G is a group, 3(G) reduces to the group

of inner automorphisms of G and Bruck has defined 3(G) so that

it will play this role throughout the theory. There are many

results in [5 ] which indicate the intimate relationship between

the properties of G and the properties of M&) •

DEFINITION 2 .1.3 : An isotopism from the quasigroup (H, • )

to the quasigroup (K,o) is an ordered triple (o^P^y) where

CK, p, and y are bisections from H onto K and (x *y)v = xa* yp for all x,y € H ; in this case (H, •) and (K, °) are said to be isotopic. An isotopism (a,p,I^) from (H,*) onto

(H, o) is called a principal isotopism and in this case (H,•) and (H,0 ) are said to be principal isotopes.

Note that "isotopic to" and "principally isotopic to" are equivalence relations. Part of the relationship between these two concepts is expressed in the following proposition.

PROPOSITION 2.1.4: If (G, •) and (H,•) are isotopic then there exists a quasigroup (G, *) such that (G, *) is a principal isotope of (G, •) and (G,*) is isomorphic to (H,») . 8^

Proof: Let (<^,p,y) "be an isotopism from (G, •) onto

(H, ° ) and define the operation * on G by x *y =

(xyoT1 ) • (yVp 1) . Then (yaT1, yp"1,^) is the desired

principal isotopism from (G,*) onto (G,*) and since for all

x',y* € G, we have (x1 * y ' ) v = x ’a o yfp , we see that for any

x,y € G, by taking x' = x y a "1 and y' = y yp -1 we obtain

(xya ^ *yyp ^)y = x yot e. yyp ^p = xy a yy . Hence, y is an

isomorphism from (G, *) onto (H,°) . □

In addition to its intrinsic interest, the study of

isotopism is important in the study of nets and also arises naturally in the study of Latin Squares. In fact, we can see that when a Latin Square is viewed as the multiplication table of quasigroup, principal isotopism amounts to permuting the rows and columns of the square. A question often asked by algebraists is the following: given a loop (G, •) with certain properties, is it true that every principal isotope of (G,•) is isomorphic to (G, •) ? This problem in algebra leads naturally to the concept of autotopism which is another very important tool in the analysis of i.p. loops.

DEFINITION 2.1.5: An ordered triple (U,V,W) is an autotopism of the i.p. loop (G, *) iff U, V, and W are permutations of G satisfying xU • yV = (x *y)W for all x,y € G . An element u € G is a Moufang element iff (ux)(yu) = [u(xy)]u for ai 1 x,y € G and the set of all Moufang elements of G is 8?

denoted by 971(G) . A permutation S of G is called a

pseudo-automorphism of G with companion c iff for al 1

x,y € G, (xS)(yS *c) = (xy)S -c . The group of all pseudo-

automorphisms of G will be denoted by Pseu-Aut(G) .

We note that G is an 771 loop iff G *= 97l(G) , the set of

autotopisms of G forms a group called Top(G) under the

coordinatewise composition law (U,V,W) • (U'^V'^W1 ) = (UU*, W*,WW*) ,

a pseudo-automorphism with companion e is an automorphism of

G, eS = e and SJ = JS for all S € Pseu-Aut(G) f u € #|(G)

iff (L(u),R(u),F( u )) € Top(G) , and if S € Pseu-Aut(G) then

CS,SR(c),SR(c)) € Top(G) . These last types of autotopisms can

be thought of as "Moufang autotopisms" and "pseudo-automorphism autotopisms" respectively.

The following crucial result gives a method of producing new autotopisms from a given one and clarifies the importance of the Moufang elements: every autotopism is the composition of a "pseudo-automorphism autotopism" with a "Moufang autotopism."

THEOREM 2.1.6: If (U,V,W) € Top(G) and u = eU and v = eV then (W,JVJ7U) € Top(G), (JUJ,W,V) € Top(G), u £ ^(G) , and there exists S € Pseu-Aut(G) such that S has companion c = vu'1 and (U,V,W) * (S,SR(c),SR(c) ) • (L(u),R(u),P(u) ) .

We begin to see the special status of commutative loops and of exp-3 loops in the following. 86

THEOREM 2*1.7: !0l(G) is a subloop of G and if

u,v € #j(G) then R(u,v) = L(u"\v is a pseudo-automorphism

of G with companion (u,v) and T(u) is a pseudo-automorphism -3 of G with companion u .

DEFINITION 2.1.8: A subloop H of G is said to be a characteristic subloop of G iff H® s H for all 0 € Aut(G) .

PROPOSITION g.1.9: If S £ Pseu-Aut(G) then S|N^

(the restriction of S to N(G)) is an automorphism of

N(G) and N(G) is a normal characteristic subloop of 77{(G) •

If G is commutative then Pseu-Aut(G) s Aut(G) .

To give a slight flavor of this material we include a proof of the following theorem. Note that Proposition I.5 .1U is a consequence *

THEOREM 2.1.10; The following three statements are equivalent for the i.p. loop (G, •) :

(i) (xy)(zx) = [x(yz)]x for all x,y,z 6 G.

(ii) [ (xy)z]y = x[ y(zy)] for all x,y,z f G.

(iii) x[y(xz)] = [(xy)x]z for all x,y,z £ G.

Furthermore, statement (i) implies that for all x,y £ G

(xx)y - x(xy), (xy)x = x(yx), and (yx)x = y(xx) .

Proof: First assume (i). By taking x = x', y = e, and z = y' we see that x'fy'x1) = fx'y'Jx' for all xf,y' 6 G. Now by Theorem 2,1.6, (L(x),R(x),L(x)R(x)) £ Top(G) implies

that (JL(x)J,L(x)R(x),R(x)) £ Top(G) so y * (xzx) = (yx)JL(x)j •

zLfx)R(x) = [ (yx) «z]R(x) = [ (yx) • z]x which is equivalent to •

(ii). From (ii) with z = e , x - y ', and y = x ’ we obtain

(y’x'Jx* ss y'fx'x') for all x',y' g G . Second we assume (ii).

Given x,y,z 6 G we have [ (z"1x-1 )y-1 ]x"*1 = z’^tx^fy'^x”1)] ;

since G has the i.p., x[y(xz)] = [(xy)x]z which is (iii). * (Similarly (iii) emplies (ii).) Now from (iii) with y = e,

x = x 1, and z = y r we obtain x'fx'y') = (x'x')y' for ai1

x*,y' 6 G . All that remains to show is that (ii) implies (i).

Assuming (ii) and taking x ’ = xy-1 we obtain (xz)y =

[(x’y)z]y = (xy"1 )[y(zy)] which is to say (xz)R(y) = xR(y_1) •

zR(y)L(y) . Hence, (R(y-1 ),R(y)L(y),R(y)) 6 Top(G) which implies

by Theorem 2.1.6 that y € #?(G) and since y was arbitrary,

7n{G) = G which is (i) , □

We remark that the theorem in Bruck*s book is actually much

stronger in that he derives from each of (i), (ii), and (iii)

that G is an i.p, loop rather than assuming it.

DEFINITION 2.1.11: For an 771 loop (G, •), 0: G -> G is a

semi -endomorph! sm of G iff for all x,y £ G (xyx)® - x®y®x®

□ and e = e , and a semi-endomorphism 0 is called a semi- automorphism of G iff 9 is bijective. The group of all

semi - e ndomorph i sms of G is written Semi-End(G) and the group of ai i semi-automorphisms of G is written Semi-Aut(G) . Note 88

that because of1 Theorem 2.1,10, expressions of the form xyx —1 6 A —1 need not be bracketed. Also note that (x ) = (x ) for

all x € G and for all @ € Semi-End(G) .

PROPOSITION 2.1.12; If G is an loop then J(G) £

Pseu-Aut(G) ^ Semi-Aut(G) . Secondly, assuming that G is

a c-% loop we have u(G) s Pseu-Aut(G) = Aut(G) and for 2 0 € Semi-End(G) the mapping x -+■ (x0 ) is an endomorphism of

G and lastly q £ End(G) iff for all x € G, o(x) ^ 2 .

DEFINITION 2.1.13: For (G, •) a loop and A,B,C £ G

[A,B,C] = {(a,b,c): a £ A, b 6 B, and c € Cj . The adjoint

of A is given by A^ = [x € G: [A,[x),G] = [e}} and the — *#■ -X- closure of A is given by _A_ = (A ) .

PROPOSITION 2.1.It; If (G, •) is an loop and A is a _ * _ non-empty subset of G then A £ A, A and A are subloops

of G and [A,A,GJ = (e) implies that [A,A,G] = {e) .

Proposition 2.1.It together with other lemmas of a

technical nature yields the following major theorem.

THEOREM 2.1.15; Assuming that (G, •) is an ^ loop and that A, B, and C are non-empty subsets of G, if

[A,A,G] = [B,B,G] = [C, C,G] = [A,B, C] = (e) then there exists an associative subloop (i.e. a subgroup) (H, *) of (G, • ) with

A U B U C £ H. 89

An immediate consequence is the following generalized

version of Moufang's Theorem.

THEOREM 2.1.16: If G is an ^ loop and a,b,c € G

associate, that is a(bc) = (ab)c , then a, b, and c generate

a subgroup of G .

COROLLARY 2.1.17 (Moufang) : Every 7ft loop is diassociative.

Proof: Since by Theorem 2.1.10, x(xy) = (xx)y in an 7ft

loop, the subloop generated by x and y is associative.

We remark that the property of an HTS, S , that every

triangle {e,a,b] generates a plane isomorphic to AG(2 ,3 )

corresponds precisely to the fact that a and b generate a

subgroup (necessarily isomorphic to C^) of the o.~7ft exp-3 loop

Ge(S) . Also, distinct non-identity elements a, b, and c

which associate in the loop Ge (3) must generate a subgroup

and if this group is isomorphic to (rather than just

C^) then by Theorem 1 .5 .2 0 , {e,a,b,c} generates a box of S .

DEFINITION 2.1.18; For an 7ft loop (G, *) the core of G .

denoted by (G. + ) , is given by the operation x + y = x • y • x .

Although I have not seen it mentioned in the literature,

I am sure that Bruck at some point became aware of the following proposition. 90

PROPOSITION 2.1.19: The core of a c-77? exp-3 loop is the

quasigroup operation of the corresponding STS.

Proof: In this case x + y = xy ^x = xxy-^" = x”^y”^

x ° y . □

So we see that just as Hall originally had an HTS without

noting the connection with loops, Bruck had the HTS’s at hand

in the context of loop theory but was apparently unaware of

their great geometric signifigance. Bruck does point out the

meaning of the core in terms of the 3 -net corresponding to the

loop (G, *) (see §5 of Chapter I). Given i-lines = (i,x)

and = (i,y), since G is an JP? loop we have the well defined

reflection, + 1^, of in ; and we see from Figure 5

that necessarily + 1^ = (i,x + y) .

x

y

Figure ?: Reflection and the Core

It is easily verified that (x + y) * z = x • z + y • z , z ♦ (x + y) = z *x + z • y , and (x + y )-1 = x"1 + y-^ . The 91

following result includes Proposition I.5.19 and also points

out that the core is invariant under principal isotopy.

THEOREM 2.1.20: If . (G, «) is an 771 loop then J»(G) = Aut((G,+));

Semi-End(G) = fe €End((G, + )): e© = ejj L(xyx) = L(x)L(y )L(x),

R(xyx) = R(x)R(y)R(x), and P(xyx) = p(x)p(y)p(x)j and if

H = (G,*) is a principal isotope of (G, •) then 1^: (G, + ) -► (H, + ) is an isomorphism.

PROPOSITION 2.1.21; If G is a diassociative loop then the following identities are equivalent:

(t-1 )\ (xyx) ( ^2 - x 2y 2 x 2 .

(ii) x(y-1xy) = (y-1xy)x .

(iii) ((x,y),x) = e .

(iv) (xn,y) = (x,y)n for all integers n.

(v) (xy)n = xnyn(x,y)-n^n ” ‘*‘^ 2 for all integers n.

(vi) (xyx)n = xnynxn for all integers n .

PROPOSITION £.1.22; If G is an ^ loop then

(i) R(x_1,y-1) == L(x,y) = L(y,x)-1 .

(ii) L(x,y) = L(xy,y) = L(x,yx) .

(iii) L(x-1,y-1 )L(x-^",y) «* L((x,y),y) .

(iv) xL(z,y) =x(x,y,2 )-1 .

(v) (x,y,z) - (x,yz,z) = (x,y,zy) .

(vi) (x,y,z) = ( xy^y) - 1 . 92

(vii) (x,y,z) = (x,y,zx) .

(viii) y[x(x,y,z)-1] = (yx)(y,x,z) .

Note that by parts (i) and (iv), L(z,y): x ►» x(x,y, z )-1 and

R(z,y): x h. x(x,y-1 ,z-1 )"1 .

These last two propositions are used to prove the following

theorem which includes the properties of associators that will

be of primary importance throughout the remainder of this

treatise.

THEOREM 2.1.23 : If G is an 77? loop then the following

seven identities are equivalent:

(i) ((x,y,z),x) = e .

(ii) (x>Y> (y,z )) = e .

(iii) (x,y,z)-1 = (x'^y^z) . , v-1 f -1 -1 -Iv (iv) (x,y,z) = (x ,y ,z ) .

(v) (x,y,z) = (x,zy,z) .

(vi) (x,y, z) = (x, z,y-1) .

(vii) (x,y,z) = (x,xy,z) .

Furthermore, each of the above implies the following identities:

1. (x,y,z) £ Z(<{x,y,z})) .

2 . (x,y,z) = (y,z,x) = (y,x,z)-1,

3 . (xn,y,z) = (x,y,z)n for all integers n .

(xy,z) = (x,z)((x,z),y)(y,z)(x,y,z)3 . 93

Notice that the last identity indicates an intimate

relationship between commutativity, associativity, and the

exponent 3 * Ik particular we have the next result in the case

that G is commutative. First we give the definition of a

most important object.

DEFINITION 2,1.2b; If G is a loop the associator subloop

of G is denoted by G_^_ and is defined by

G ’ = <[(x,y,z): x,y,z € G}) , i.e. G* is the subloop of G

generated by all the associators of G . The cubing map is

given by C: G -*■ G_ and we write Tn = (x € G: x3 = e] . x - x3 —

FROPOSITION 2.1.25: If G is a c-^ loop then

C € End(G), x3 = C(x) e Z(G) for all x € G, and G ’ = TQ .

Notice that this is stating that the associator subloop of any c-7!\ loop is a c-ty exp-3 loop. Next we see the last aspect of the special role of the number 3 in the study of % loops.

THEOREM £.1.26: If G is an % loop and if the mapping 2 x x is a semi-automorphism of G then T„ is a normal

O subloop of G, the mapping N(G)x ►* N(G)x is an endomorphism of G/N(G) , and for all x € G NfGjx3 € Z(G/N(G)) . 9 k

§ 2. Commutative Moufang Loops

In this section we shall point out how the results of §1 of this chapter can be applied to the study of cloops.

Particularly important will be the properties of associators which will be used constantly in our upcoming calculations.

We shall then discuss briefly the rudiments of Bruckrs theory of nilpotence as it will be applied to the special case of central nilpotence. Lastly we shall summarize part of Chapter

VIII of Bruck1s book [6 ] in much the same way that we have just summarized Chapter VII. As mentioned earlier, much of this work is pointed toward proving the Bruck-Slaby Theorem.

There are quite a number of results which are used to prove this theorem but then become immediate consequences of it once it has been proved. We shall omit such a result from our summary except when it will be used extensively throughout the remainder of this work.

We recall that by Proposition 2,1.12, if G is a c-ty loop then JT(G) ^ Aut(G) (however we see that Moufang's

Theorem implies that T(x) = IQ for all x € G) ; also in the case that G is a c-%1 3-loop Proposition 2.1.12 allows us to conclude that Semi-End(G) = End(G) . And the following important theorem is an immediate consequence of Theorem 2.1.23.

THEOREM 2.2.1; If G is a c-W{ loop then the following identities hold in G : 95

(i) (x,y,z) = (x,xy,z) .

(ii) (x,y,z) = (y,z,x) = (y,x,z)- 1 .

(iii) (xn,y,z) = (x,y,z)n for all integers n.

(iv) (x,y,z)3 = e .

Furthermore,

(v) (x,y,z) e Z«{x,y,z) >) for all x,y,z € G .

Note that (ii) is stating that if the three arguments of an

associator are permuted by an even permutation then the value

of the associator is unchanged and if the arguments are permuted

by an odd permutation then the new value of the associator is the inverse of the original value. Also note that for n = -1

in (iii) we obtain (x \y,z) = (x,y,z)~^ and that (iv) has already appeared in the conclusion of Proposition 2.1.25 .

DEFINITION 2.2.2; Let (G, •) be a c-Wf loop. For subsets

A,B,C £ G we write (A.B.C) = ({(a^b^c): a e A, b € B, and c € C)) and we define inductively the lower central series,

GQ,G^Gg,.. • f of G by G^ = G and = (G^ ^G^G) for i > 1 . Note that G^ = G' , the associator subloop of G .

To define the upper central series, Z^fG), Z^(G), ... , of

G we set (G) = (e} and for i > 1, Z^G) is the inverse image of Z(G/Z^ ^(G)) under the canonical mapping

G -*• G/Z± -;l(G) . Notice that Z^G) = Z(G) , G £ G± _ ; and x m . Z± _ x(G)x

Z^(G) 3 Z± „ j(G) • G is said to be lower centrally nilpotent 96

iff for some i', G^f = (e) and upper centrally niipotent iff

for some i", Z^„(G) = G . If G is lower centrally niipotent

then the lower nilpotence class of G is defined by

nil(G) = min {iT : G^, = [e)J and if G is upper centrally

niipotent then the upper nilpotence class of G is defined by

nil(G) « min {i": Z±t1(G) = G] .

A very general theorem of Bruck [5] can be applied to

central nilpotence in c-17[ loops to obtain the following result.

THEOREM 2.2.3; Assume that G is a c-ty loop; then G

is lower centrally niipotent with nil(G) = n iff G is upper centrally niipotent with nil(G) = n .

By virtue of this theorem we may just refer to a lower

centrally niipotent c-7/l loop G as being niipotent and to nil(G) as its nilpotence class. Note that nil(G) = 1 iff G is an abelian group. We shall see that c-771 loops G with nil(G) = 2 are the easiest non-associative cloops with which to deal and that the structure of c-77\ loops becomes more

complex in the cases of larger nilpotence classes. We recall

Definition 2.1,13 and note that if nil(G) = 2 then G^ =

(G'jGjG) *= {e} means that G' £ 2(G) so that (G')* = G and

G7 = Z(G) .

We shall now return to our summary of the material in

Bruck's book. We assume throughout that G is a c-77J loop. 9 7

As mentioned earlier, the following theorem which is known as

the Bruck-Slaby Theorem is the culmination of a huge amount of

work by Bruck. Slaby's name appears because he collaborated

with Bruck for the cases d = 1* and d = 5 and formulated

the statement of the theorem.

THEOREM 2,2.1+ (Bruck and Slaby); If d is a positive

integer with d > 3 a^id if G I s a c-7% loop which is generated

by d elements then G is niipotent and nil(G) < d - 1 .

As a corollary we note the validity of Theorem 1.5*17 *

Suppose that for a finitely generated c-J5? loop G we had

Z(G) = te} . Then by definition we would have {e} = zo(G) =

Z1 (G) = Z2 (G) = ... which by Theorem 2 ,2.3 would contradict

the nilpotence of G which is insured by Theorem 2.2,1* .

Hence Z(G) must be non-trivial.

DEFINITION 2.2.5; For d 6 N we denote by F^ the free

c-/7? loop on d generators and we shall write k(d) = nil() ,

We shall also write c(d) = 1 + [d/2] where [x] denotes the

largest integer not exceeding x .

We note that if f: G -► H i s a surjective homomorphism of

c-J5? loops then since (x,y,z)^ = (x^, y^, z^) we must have

G^ = for 0 < i < nil(G) and therefore nil(H) < nil(G) *

Hence, if one is able to construct any c-fy loop G on d 98

generators then automatically k(d) > nil(G) . By a very

clever construction which will be analyzed in some detail in

Chapter III, Bruck finds for each d > 3 a c-fy loop such

that nil(B^) = c(d) . Therefore, the situation at that time

was that c(d) < k(d) < d - 1 . The smallest value of d

for which c(d)

prove the following.

PROPOSITION 2.2.6: If k(5) = c(5) = 3 then k(d) = c(d)

for all d > 3 .

We shall discuss in §3 of this chapter the recent work which has shown that the conclusion of the above proposition is not the case, and in fact k(d) = d - 1 .

The next result is obtained by studying the structure of c-7% loops generated by four elements and is the most widely used technical tool in this treatise,

THEOREM 2,£.7: If w, x, y, and z are elements of the c-77} loop G then

(wx,y,z) = (w,y,z)(x,y,z)[ ((w,y,z),w,x)((x,y,z),x,w)] .

This identity is called the expansion law and when it is used in computations we shall refer to it as (E) , In some situations

(for example, when nil(G) =2) the expansion law simplifies to 99

(wx,y,z) = (w,y,z)(x,y, z ) . We shall call this identity

simplified expansion and refer to it as (SE) .

The next two propositions are used to prove the theorem

that follows.

PROPOSITION 2.2.8; Assume that H is a subloop of the

cloop G and that a,b € G . If for all x 6 G, (x,a,b) € H

then also ( (a,b,x1 ), ) £ H, (((a,b, x^ x ^ X g J^x^x^ ) € H, ...

for all elements x^x^x^, ... of G.

PROPOSITION 2.2.9: Assume that G is a c-/7j loop and

a,b,c,d 6 G . If for all x € G, ((x,a,b), c,d) = e then

e = (C(a,b,x1 ),x1 ,x2 ),c,d) *= ((((a,b,x1 ),x1,x2 ),3(g,x3 ),c,d) = ...

for all elements x^x^jX^,... of G.

THEOREM 2.2.10: Assume that G is a c-ty loop and that

G is generated by the subset r £ G (we write G = (r)) and

let x be an element of G . Then x € Z^G) iff

((...((xjV^Yg^YyY^).*. ),Ya _ i^ 21) = e for all elements

Y1 ,Y2, • ..,V2i € T. Also, G± = £e) iff

((■•■((Y1 ,Y2 ,Y3 ),YI+,Y5 )*-.),Y2 ijY2i + 1) = e for all elements

Y1 'Y2 * * “ ** Y2 i + 1 € r *

In the next section we discuss a conjecture called the

Triple Argument Hypothesis. The first identity in the following proposition suggests that this conjecture is plausible. The second identity will be most important in Chapter VI when

we investigate the structure of c-/7f loops of nilpotence class

3 on four generators.

PROPOSITION 2.2.H: The elements v, v, x, y, and z of

a c-% loop satisfy the following identities:

(i) (v, (v,w,x); (v,y,z)) = e .

(ii) C(x,y,w),w,z) = ((y,z,w),w,x) .

At this point Bruck introduces a calculus of associator

subloops which resembles Philip Hall's calculus of commutator

subgroups of a group. It yields some very important properties

of the upper and lower central series of c-7?i loops.

DEFINITION 2.2.12: The derived series G^°\g^^,G^2^, ...

of the c-7% loop G is defined by G^°^ = G and G ^ + ^ =

( G ^ ) ' = ( G ^ ^ G ^ ^ G ^ ) . For i < 0 we define Z (G) = te) .

THEOREM 2.2.13: If G is a c-7% loop then

i + j + k + 1 * (ii) (Gi,GJ,Zk(G)) £ Z

(iii) G ^ c G . (3-l)/2

Notice in particular that if nil(G) < 3 then (G',G',G) = (e) and that if nil(G) < k, G ^ = ( G ^ G ^ G ’) * te) ; that is, the associator subloop G' is a group. 101

PROPOSITI OH 2.2.14; Assume that G is a c loop. If

a given element d € G satisfies any one of the following

equalities for all w,x,y,z € G then d satisfies each of

these equalities for all w^Xjy, z € G :

U) (xy,z,d) = (xj,z,d)(y,z,d) • (ii) (xd,y,z) = (xj,y,z)(d,y,z) .

(iii) ((w^y^Zjd) = e .

(iv) ((d,w,x),y,z) = ((d,y,x),w,z)_1 .

(v) ((x,y,z),x,d) = e .

(vi) ((d,x,y),x,z) = e .

DEFINITION 2.2.15: The distributor of G is defined by

D(G) = {d € G: (xy,z,d) = (x,z,d)(y,z,d) for all x,y,z £ G) .

The lower distributor series Lo(G),L^(G),__ of G is defined inductively by L0(G) = G and Li + 1(G) = (L^G^G*) for all non-negative integers i .

PROPOSITION 2.2.16: If G is a c-ty loop and d £ S then D(G) is a characteristic normal subloop of G , 1 ^ = 0 ^ ,

c G^^ for all non-negative integers i , ^(G) £ D(G) , and if G is generated by d elements then D(G) c zc^j(G) and for all non-negative integers if G2 + ic(d) C Li + 1^G^ *

Note that by observing (i) of Proposition 2,2.14 we see that D(G) is the set of elements whose presence as an argument in an associator admits the simplified expansion law for that ice

associator. Furthermore, by Theorem 2.2.13 and Preposition

2.2.1k- we see that if nil(G) = n then G „ c D(G) . n - d.

PROPOSITION 2.2.17; If G Is a c-/7| loop generated by

d > 3 elements and satisfies (G,G',Gf) = (e) then = •

This result is the best possible since the Bruck loops

Bd to be studied in Chapter III satisfy (B^,= (e) and

(Bd)i / (©} for i < c(d) .

The next proposition provides us with the basic five

element identities.

PROPOSITION 2.2.18: If G is a c-ty loop and v,w,x,y, z £ G

then

(i) ((v,y,z),(v,y,z),x) = ((w,y,z), (x,y,z),v)

(ii) (((v,y,z),w,x),y,z) = ((v,y,z), (w,y,z),x)_1

(iii) ((v,w,x),y,z) = ((v,y,z),v,x)((v,y,z),x,v)((x,y,z),v,v)

(iv) ((v,w,x),y,z) = ((y,w,x),v,z)((v,y,x),w,z)((v,w,y),x,z)

(v) (((x,y,v),v,w),w,z) = (((z,x,w),w,v),v,y)

(vi) ((x,y,vw),vw,z) - ((x,y,v),v,z)((x,y,w),w,z)

(((x,y,v),v,w),w,z)((x,y,v),w,z)

C(x, y,w),v,z)

Bruck proves many more identities and structure theorems, but we shall conclude our partial summary with the following results. 103

PR0P0SITION 2.2.19; Assume that G is a c-27? loop and

that k is a non-negative integer. If x,y € G satisfy

(G, {x],ty}) c Zfc(G) then (G^ £x),(y)) for all

non-negative integers i . Furthermore, if y' £ G also

satisfies (G,£x},{y')) £ Z^(G) then ((G±,Gj,(x)),[y},(yf)) c

Zk _i *>or non-negative integers i and j .

§ 3 • Mev Contributions to the Theory

In addition to summarizing the recent work of others we include in this section the original construction of Theorem

2.3.18 . We begin with a summary of the 1977 work of Lucien

Beneteau [2]. Much of it, in particular the analogue of the

Burnside Basis Theorem and the results involving the Frattini subloop of a c~77i loop, had already been done by Bruck in [5 ] ,

We shall only occasionly include a proof here. All c-JT? loops discussed in this section are assumed to be finitely generated.

DEFINITION 2.3.1? For a c-57; loop G , the Frattini subloop of G is defined by $(G) = Pi fM: M is a maximal proper subloop of G) . B £ G is a basis of G iff B is a minimal generating set of G and the set of all bases of G is written fl(G) . If any basis of G is finite then G is said to be finitely generated.

An element x € G is a non-generator of G iff (tx) U S) = G implies that (S) = G . For subsets A,B £ G AB denotes

{ab: a 6 A and H B) . For a prime p , a p-group is a group I C k

in which every non-identity element has order pm for some

m > 1 * A p-group is said to "be elementary iff every non-identity

element has order p .

PROPOSITION 2.3.2; #(G) = (x 6 G: x is a non-generator of G)

and is a normal subloop of G .

The following result is crucial to our study of the

structure of c-/7f exp-3 loops, so we include its proof.

PROPOSITION 2 .3.3: If H is a normal subloop of the c-27?

loop G then G/H is a group iff H 2 G' .

Proof: First assume that H a G' . Given Hx,Hy,Hz € G/H

we have (Hx • Hy) • Hz = H[(xy)z] = H[ (x,y,z)[x(yz)] ] =

(H(x,y,z) )[x(yz)] = H[x(yz)] since (x,y,z) € G 1 Continuing, * we have H[x(yz)] = Hx • (Hy *Hz) . Hence G/H is associative

and therefore is a group, (Note that a quotient loop of a c-7!\

loop is always a c-57f loop as well. ) Next assume that G/H is

a group. Given an associator (x,y, z) € G r we see that

H[x(yz)] »H(x,y,z) = H[ (xy)z] = H[x(yz)] , the last equation

following from the fact that G/H is a group. Therefore,

H(x,y,z) = H forcing (x,y,z) € H . Hence, G* £ H . □

PROPOSITION 2 . 3 A ; If M is a maximal proper subloop of the finitely generated c-ty loop G then M A G and G/m is a cyclic group of prime order. 105

PROPOSITION 2.3.5 1 With G -+ G/$(G) being the canonical x $(G)x

quotient mapping, B 6 fl(G) iff £ B(G/$(G)) j furthermore,

B| . |B^I .

THEOREM 2.3.6: Assume that G is a finitely generated

c-7H 3-loop. Then

(i) if M is a maximal proper subloop of G then [G:M] = 3 .

(ii) $(G) = G ’F where F = {x^: x € G) .

(iii) G/$(G) is an elementary abelian 3-group.

COROLLARY 2.3.7: If G is a finitely generated exp-3 loop then $(G) = G' .

By using Proposition 2.3.5 in conjunction with Theorem 2,3.6 and Corollary 2 .3.7 one obtains the loop analogue of the Burnside

Basis Theorem for p-groups. We state it only for the case p = 3 which in light of Theorem 2.1.1 is all that is necessary.

THEOREM 2.3.8: If G is a finitely generated c-ty 3-loop then every basis B € B(G) has the same cardinality. If G is an exp-3 c-7?? loop then this cardinality is equal to the number of generators of the elementary abelian 3-group G/G' and in fact, B € 13(G) iff B^ € B(G/G’ ) . 106

DEFINITION 2.3*91 We denote the cardinality of any basis

of the finitely generated c-2?f 3-loop G by d(G) and call this

number the dimension of G .

The next two results are used constantly in this treatise,

so we include their proofs.

THEOREM 2.3.10? If G is a finitely generated c-JT; 3-loop

with dimension d(G) = d then for all i with 1 < i < nil(G),

G . A G (and so G. A G. .) and G. ,/G. is an abelian i — ^ i — i -1 7 i-l/ i 3-group. Furthermore G is finite.

- Proof: By Theorem 2.2.U the lower central series of G

is given by G = Gq 2 G^ 2 ... 2 Gnil^Gj = (e) where

nil(G) < d - 1 . Fix i with 1 < i < nil(G) ; given g 6 G.

we know that for x,y £ G the associator (g,x,y) £ + i c *

Hence, the element (gx)y £ (G^xjy and also (gx)y =

(g,x,y)[g(xy)] « ([ (g,x,y)g][xy] ) 6 G.(xy) , the second equality

following from the fact that (g,x,y) £ Z(({g,x,y})) which is part of Theorem 2.2.1 . Therefore (G^x)y = G^(xy) and so

G^ A G and a fortiori G^ A ^ _ i * Next G^ = (G^ _ ^,G,G) 2

(Gi _ _ j/G'i _ l) = G^ _ 1 so by Proposition 2.3.3, we see that G^ 1/Gi is a group and so must be an abelian 3-group.

(Note that G^ ^ G^ ^ since i - 1 < nil(G) and in the case that G has exponent 3> l ^ i 811 elemen^ary abelian 3-group.) The finiteness of G is proved by induction on nil(G) . If nil(G) *» 1 then G is a finitely generated abelian 3-group which by the Fundamental Theorem of finitely generated abelian groups is the direct product of d(G) cyclic 3-groups, each of which is finite; hence, G is finite. Assume that nil(G) = n > 1 and that all finitely generated c-V\ 3-loops of nilpotence class less than n are finite; we consider G' and g/g ' .

G/G' is an abelian 3-group and by Proposition 2.3*5 and Theorem

2.3*6, d(G/G') < d(G) ; hence G/G* is finite by the same argument used above for the case of nil(G) = 1 . Wow for all i satisfying 1 < i < nil(G) - 1 we have G^ = (G ^ G ^ G ^ s

(G.,G,G) = G, . ; and in particular, we have G* , ^ l l+l * ’ n -1 Gn = £e} . Thus nil(G') < nil(G) , so by induction G 1 is finite. Hence G is finite and |g| = |g/G* | • |g'| . □

We remark that Bruck actually proved the more general result that any finitely generated c-7% loop is finite.

COROLLARY 2.3.11; If G is a c-^ 3-loop of dimension d(G) then jG | = 3S^ for some positive integer s(G) .

Proof: By Theorem 2.3.10, G is finite and aii the abelian

3-groups Gi _ are finite and have cardinalities which are nil(G) , powers of 3* Hence |g | = TT |G. /G. | = 3 * □ i = 1 1 ' x 1

PROPOSITION 2.3.12 fBeneteau); If G is a c-J7j 3-loop then s(G) > d(G) and G is a group iff s(G) « d(G) . 108

The next result is a ma^or fact about HTS's which was used

in the proof of Proposition 1.4.21 and which will be of great

use in the study of HTS's in the upcoming chapters.

COROLLARY 2.3.13: If S = (X,fi) is an HTS then |x| = 3m

for some m > 4 .

Proof: By Theorem 1.5.16 and Corollary I.5.21 the loop

Ge(S) is a c-7l[ exp-3 loop which is not a group. By Corollary

2 .3. H and Proposition 2.3.12, |X| = 3™ where m = s(Ge(S)) >

d(G€ (S)) . By Moufang's Theorem (Corollary 2.1.17) d(G ©(S)) > 2 and so m > 4 . □

PROPOSITION 2.3.14 (Beneteau): If G is a finitely

generated non-associative c-ty loop then [G:Z(G)] > 3 .

COROLLARY 2.3.15 (Beneteau): If G is a finitely generated non-associative c-ty 3-loop then 3^ < (2(G)| < 3S^ .

The lower bound follows from the upper central nilpotence of G and the upper bound is a consequence of the fact that x^ € Z(G) for all x € G (see Proposition 2.1.25) which means that G/Z(G) has exponent 3 and the fact that G would be a group if d(G/Z(G)) < 2 . Note that sfGg^ = 4 and so |z(Gg1)| = 3 , a fact used in the proof of Theorem 1.6.2 . 109

PROPOSITION 2.3*16 (Beneteau): If G is a finitely

generated c3-loop and d(G/Z(G)) < d(G) then |z(G) | > 32 .

Before giving Beneteau's final contributions we pause to

give a construction. First we state the following result whose'

proof is immediate from the definition of the direct product

of loops,

PROPOSITION 2.3.17: If G and H are c-ty loops then

Z(G ® H) = Z(G) « Z(H), (G ® H)j_ « G± ® H± i and d(G ® H) =

d(G) + d(H) .

Now in [67] Young constructed for each d > 3 a c-??j d +1 exp-3 loop, Y^, of cardinality 3 by taking Y^ =

GQl ® C - 3 . Note that d(Yfl) - 3 + (d - 3) = d, |Y^| = 3, J ^ J Q and |z(Yd ) | = 3 * 3 = 3 ~ which is as large as possible

by Corollary 2.3.15 . We generalize his construction as follows.

THEOREM 2.3.18: For any positive integer and any ml “2 non-negative integer nu,, Gg^ ® Cg , which we denote by

RfnLj^m^) , is a non-associative c-ty exp-3 loop satisfying +nu iil l^(lnl,in2) I = 3 ' “ 3 , and |Z(RCin^,)) | <=

J • 110 . nu nu 4m_ +nu Proof: iRtm^mg) | = (3 ) -3 = 3 . .

E ’C^,^) = (G^) ■L ® (e) . Now dCG^/G^) = dfG^) = 3,

e. so |Gq^ | - 3 and hence |R1 ) | = 3 . Since > 1

Rfm^iiig) is non-associative and in fact nilfRfnL^m^)) = 2 ,

m_. iru Lastly, Z t R ^ , ^ ) ) = Z(G8l> ® C3 so ^(ROn^n^)) | =

hl +eu 3 □

For the following discussion please refer to Table 1. 3™i Since [Rfm^m^): ZtRfm^m^))] = 3 we see that for roughly z hi one third of all ordered pairs (3,3) with m, z € N and

1 < z < m - 3 we have constructed a non-associative c-2??

exp-3 loop G (and hence an HTS) with |g| = 3m and

|Z(G) I es 3Z . That is, in Table 1 every third diagonal is

filled in with some R(ni^,mg) with the exception of same of the topmost entries of these diagonals. One of our main results of Chapter IV will be the construction of a loop for each place in the table with the sole exception of the entry

O 1,^) for which no such loop exists. Note that the loops

R(2,0) and R(l,^) provide us with the first example of non-isomorphic non-associative c-Tft exp-3 loops having the same cardinality. Note also that = R(l,d-3) . Ill

Table 1: R(m^,ia>)

10 11

R(2,2)

The following theorems are the main results of Beneteau's paper and were the first successful efforts in trying to catalogue the "small" HTS's since Hall had proved that was the unique

HTS of order 3 .

THEOREM 2.3.19 (Beneteau): For all d > 3f is the unique d +1 c-ty exp-3 loop having cardinality 3 and center of cardinality „d - 2

THEOREM 2.3.20 (Beneteau); = GQl « = R(l,l) is the unique non-associative c-17l exp-3 loop of cardinality y .

In Chapter V we shall give a better proof of this theorem than

Beneteau's in that It provides much greater insight into why there 112 % is no c-771 exp-3 loop G with |G| = 3^ end |Z(G)| - 31 .

As we have already mentioned, Bruck was extremely interested

in finding k(d) , the nilpotence class of the free cloop on

d generators. In fact, he posed this problem in a special article

[7]. Bruckrs original hope was that an interesting associator

calculus for c-ty loops would develop which would be similar to

the commutator calculus of group theory (see [i+0], for example).

We have seen that Bruck himself began such a calculus, particularly

dealing with identities involving five or fewer elements. However, he was worried that perhaps k(5 ) = 3 and then k(d) = c(d) for all d > 5 • If this were to be the case then the associator calculus could be dealt with wholly in terms of exterior algebras since Bruck's loops B^ , to be described in Chapter III, satisfying d(B^) = d and ni^B^) - c(d) are constructed directly from the exterior algebra over GF(3) , (See [59] for a description of the exterior algebra over an arbitrary field.) In November of

1978> E. H. Smith greatly extended the associator calculus in his article [5 b]. Smith then used this extremely complicated work in conjunction with techniques of niipotent group theory (in studying the multiplication groups of loops) to prove the following theorem (see [55]).

THEOREM 2.3.21 (Smith): For all d > 2, k(d) = d - 1 .

So apparently Bruck's original belief that the associator calculus was much more rich than the study of exterior algebras 113

had been confirmed. However, in October of 197 8 Jean-Pierre

Malbos had independently proved in [^1] that k(d ) = d - l by

proving the following theorem. It is obtained by a direct and

very beautiful construction which is a variation of Bruch's

original method and involves nothing more than the exterior * algebra over GF(3) •

THEOREM 2.3-22 (Malbos): For all d > 2 there exists a

c-771 exp-3 loop denoted by which satisfies d(Md) = d and

nil(Md) - d - 1 .

This again suggested that the associator calculus entailed

only the study of exterior algebras but Smith apparently has

shown in [56] that this is not the case. We shall discuss

Malbos1 construction, at least briefly, in Chapter III.

Smith's proof of Theorem 2.3.21 is aided by his assuming the Triple Argument Hypothesis. This is a conjecture which states that in a Q.-7H. loop G , any associator in which an element of

G appears three times as an argument must be equal to the identity. CHAPTER III

THE BRUCK-CONSTRUCTION

§ 1. The Loops and

In this section we present an analysis of Bruch's construction of an infinite c~7% exp-3 loop 6 which contains an infinite sequence of subloops B^B^jB^j... satisfying = ^ and nil(Bd ) = c(d) for all integers d > 3 . As an original contribution we shall determine |B(jU Bd> an<^ Z (B^) . We then construct a new infinite sequence KyK^jK^... of subloops of ft satisfying B^ c Kd and nil(Kj) = c(d) . We also determine

and Z(Kd ) . Of course each for d > 2 corresponds to an HTS. For d > 3 these HTS's are new and as we have seen in

§ U of Chapter I} they can be used as PMDf s to construct other interesting designs. As we have seen in the proof of Theorem

1 .5.20 and have discussed in the second paragraph of § 6 of

Chapter I, the center of a c-ty exp-3 loop plays an important role in the corresponding HTS.

DEFINITION 3 .1.1: In this chapter we denote GF(3) by

F . For every finite subset 3 of N , including 0, we let

Xg be an indeterminate symbol. We shall abbreviate by x± , jj by , and so forth. For each non-negative

111* 115 i 1 integer i we define A_ = ( £ knxg : I g N, k e F, and n = 1 n

|Sj = 1 for 1 < n < JL) ; that is, is the set of all finite

linear combinations with coefficients in F of indeterminates

whose corresponding set has cardinality i . For each non-negative « . I integer j we define © A = { £ k x„ : l 6 N, k € F, and , n S 7 n ' i = j n = 1 n

CD j |S | > j for 1 < n < X) . We denote © A by & * fi is n i =0

a vector space over F and is endowed with the standard additive

structure; we shall denote the additive identity element of <$ — °° -I by 0 . Notice that for all i , A and © AL are also vector J = i spaces over F . We shall be particularly concerned with A^- .

We next define a multiplication on

Xg *xT = 0 if S (1 T / ^ and x Q . = (-l)p ^S,T^xg U T if

S fl T = 0 where pfS.T) = |{(s,t): s € S, t £ T, and s > t} | .

PROPOSITION 3<1»2 (Bruck): ($*•) is an associative algebra with multiplicative identity element .

Notice the following properties of the multiplication in tf : either ^ * xs = xs ' or ^ ‘ xs = -^xs ' ' in x. »x. = -x. »x, for all i.j 6 N; and if S = (i-,i0,...,i ] i j j i ’ * 1* 2f f n where i, < i_ < ... < i then x_ = x, *x. • ... «x . . 1 2 n S i, i ■ i 1 2 n

PROPOSITION 3*1*3 (Bruck): For all a,b € A1 a • b = -b • a . I Proof: With a = E k x . n n T, - k m ' x m we have

I I I tI ■ E E m = 1 n~l

DEFINITION The Bruck loop (ft, *) is defined by

fl = t(a,x): a 6 A1 and x 6

(a + b,x + y + (x-y)ab) where juxtaposition indicates multiplication in 8 .

In Bruck's book the proof of the following is not given; we include it here so that the reader can accustom himself to working in ( fl,*) ,

THEOREM fBruck): (Hi.*) is a c-7? exp-3 loop with identity element (0,0) in which (a,x)-1 = (-a,-x) .

Proof: (b,y) * (a,x) e (b + a,y + x + (y-x)ba) =

(a +b,x + y + -(x-y) • (-ab)) = (a + b,x + y + (x-y)ab) =

(a,x) * (b,y) , the second equality foU.owing from the anti- commutativity of A^" ; hence the operation * Is commutative.

(a;x) * (0,0) = (a + 0 , x + 0 + xaO) = (a,x) and (a,x) * (-a,-x) =

(0,0 + -x *a(-a)) = (0,0) since a *a = 0 . So to show that ft is a loop we need only show that is is a quasigroup. Given

(a,x),(c,z) £ fi assume that there is a solution (b,y) to the equation (a,x) * (b,y) = (c,z); then (-a,-x) * [(a,x) * (b,y)] =

(-a,-x) * (c,z) . But (-a,-x) * [ (a,x) * (b,y)] =

(-a,-x) * (a + b,x + y + (x-y)ab) = 117

(b,y + (x-y)ab + (x-y - (x - y)ab)(-a) (a + b)) =

(b,y + (x-y)ab + (x-y)(-ab)) = (b,y) ; so (-a,-x) * (c,z) e

(-a+c,-x + z + (-x-z)(-a)c) e (-a+c,-x + z + (x + z)ac) is the only possible solution. We see that it is in fact the unique

solution: (a,x) * (-a + c,-x + z + (x + z)ac) =

(c,z + (x + z)ac + (-x - z - (x+ z)ac)(a)(-a + c)) =

(c,z+(x + z)ac + (-x-z)ac) = (c,z) . Hence, fl is a loop.

Next (a,x) * (a,x) = (a + a,x + x+0) = (-a,-x) = (a,x)-^ so we see that fi has exponent 3 . Before showing that the /7j-identity is satisfied we verify that for all (a,x),(b,y) 6 B

(a,x) * t(a,x) * (b,y)] = (-a,-x) * (b,y): (a,x) * [ (a,x) * (b,y)] =

(a,x) * (a + b,x + y + (x-y)ab) -

(-a + b, -x + y + (x-y )ab + (-y - (x - y)ab)(a)(a + b)) =

(-a+b,-x + y + (x-y)(ab) - y(ab)) = (-a + b, -x + y + (x + y)ab) =

C-a + b,-x + y + (-x-y)(-a)b) = (-a,-x) * (b,y) . Now we have

(a,x) *[ («,x) *[ (b,y) * (c,z)]] = (-a,-x) * (b + c,y + z + (y - z )bc) =

(-a + b + c,-x + y + z + (y-z )bc + (-x-y-z-(y-z )bc) (-a) (b + c)) =

(-a+b + c,-x + y + 2 + (y-z )bc + (x + y + z )(ab + ac) ) . On the other hand, [ (a,x) * (b,y)] * [(ajX) * (c,z)] «

(a + b,x + y + (x-y)ab) * (a + c,x + z + (x-z)ac) =

(-a + b-t-c,-x + y + z + (x - y)ab + (x - z)ac + (y - z + (x - y)ab - (x - z )ac)

(ac-ab + bc)) = (-a + b + c,-x + y + z + (x - y)ab + (x - z)ac + (y-z)

(ac-ab + bc)) = (-a + b + c, -x + y + z + (x + y + z)(ab + ac) + (y - z)bc) .

Hence, ft is an 2»| loop. □ 118

PROPOSITION 3 • 1 • 6 (Bruck): In J3 the associator

((ai/y1 )>(a2Jy2 )>(a3#y3 )) = (0jy!a2a3 " y2aia3 + y3aia2 ^ and = ((0,0)) .

Proof : Let Xi = for 1 £ 1 5 3 • In any loop

[x(yz)](x,y,z) = (xy)z implies that (x,y,z) = (y-1z-1)(x"^(xy)z])

since (x,y,z) € Z(((x,y,z))) . Therefore, we have X^ * Xg =

(ai + a2J yi + y2 + ^yl - y2 ^aia2 ^ ’ ^X1 * X2 ^ * X3 =

Cai + a2 + a3J yi + y2 + y3 + ^yl ■ y2 )aia2 + ^yi + y2 ~ y3 + ^yl “ y2 ^aia2 ^

^ai + a2 ^a3^ = (ai + a2 + a3'yi + y2 +y3 + ^yl "y2 ^aia2 + +

Caia3 + a2a3 ^ ^ ; Xl"L * t txi * *2 ^ * X3^ = (a2 + a3^ya + y3 + fyi " y2^aia2 + (yx + y2 - y3 ) (a ^ + a^a) + {y± - y2 - y^ - (yi - y2 - (yi + y£ - y3)

(^a + a£a3 )) (-a^g - a^a )) = (a2 + a3,y2 + + (y^ - y2 Ja^g +

(y-]_ + y£ - y3 )(axa3 + a2a3 ) + (-yx + y2 + y3 )(a!a2 + aia3)) =

(a2 + &3f y2 + y3 + y3aia2 “ y2aia3 + ^yl + y2 - y3 ^a2a3 ^ ; ^ ^ = C-a2 * a3>-y2 " y3 + C-y2 + y3 )a2a3 ) » and finally (XpXg,X3 ) =

(0,y3aia2 - y2 a1a3 + (y! + y2 " y3 >a2a3 + ("y2 + y3)a2&3 + ^y2 + y3 +

(-y2 + y3 )a2a3 - y3a!a2 + y2ai&3 “ (yl + y2 ’ y3 )a2a3 ) {_a2a3 " a3a2 )> " (0,y3a1a2 ■ y2aia3 +yia2&3^ ‘ Next we see that by repeated use of the expansion law, to show that (fi1, fi1, fi) = {(0,0 )] it suffices to show that for any associators A1 = ((a1,y1 ), (a2,y2 ), (a3Jy3 )) and Ag = ((a^,y^), (a^y^ ), (a6,y6 )) and any (b,y) £ fl the element (ApAg, (b,y)) is equal to (0,0) . Now by the first part of this proposition, (ApAg, (b,y)) =

((0,yxa2a3 - y2a1a3 + y3a1a2 ), (O ^ a ^ a g - y^ a g + y6a ^ ), (b,y)) =

(0,0 +0 + 0) = (0,0) . □ 119

In particular note that since (fi* , = ((0,0)} , fl'

is a group. The following proposition will enable us to

determine the nilpotence class of the Bruck loops B^ .

PROPOSITION 3.1.7: Given n = 2m + 1 where m is a

positive integer and given elements X^ = (a^,y^) of ip for

1 < i < n, we let X = (X^Xg,.. *>^n) and define A^(X) =

(X^XgjX^) and for j with 2 < j < m, Aj(X) =

(Aj _ + 1^ * Then for J with 1 < j < m,

Aj(X) = (0,y1asa ^ a ^ ... a2j+l " ^ l ^ l ^ *** a2j +1 +

y3ala2 \ a5 *** a2j + 1^ '

Proof: By Proposition 3*1*6 the conclusion is valid for

j = 1 . Assume that the conclusion is valid for the value j -1 with 1 < j -1 < m . Then Aj(X) - (A + x) »

( ( ® ^ ^ i a2a 3ai| * * * a2j - 1 " y2 ai & 3a4 * * * &2j - 1 + y 3 ai a2 aU * * " S2j - 1 ^ *

J,y2J )' (a2j +l ,y2j +1^ ) = ( ^ ( yia2a3ah *'* a2j -1 y2a!a3ah * * * a2 j - 1 + y3aia2ah * * * a2 j - l^a2,ja2 j + 1^ =

f °'yia2a3aU * * * a2j + 1 " y2ala3a^ * * * a2j +1 + y3aia2aif * * • a2 j +1^' D

Note that because of the anti-commutativity of A^" the

Triple Argument Hypothesis is valid for the loop fi .

DEFINITION 3*1*8: For a finitely generated c-/7; loop G with fixed basis r and for 1 < i < nil(G) -1 we define inductively an i-associator as follows; a l-associator is an associator (x,y, z) where x,y,z 6 0 and for i > 1 an 120

i-associator is an associator of the form (A x,y) where i " l* x,y 6 G and is an (i - 1)-associator. An i-associator

is said to be a basic i-associator iff all its individual

arguments are elements of f. For example, ((y^, y^, )

where y^ € T is a basic 2-associator. We write BA(i) for

the set of all basic i-associators.

In this and in subsequent chapters we shall make use of the

fact that in a finitely generated loop G , the associator

nil(G) -1 subloop is generated by U BA{i) . To write (x,y,z) as i =1 a product of basic associators one writes each of x, y, and z

as a product of elements of the basis r and then makes repeated

use of the expansion law. The key fact is that the process of

expansion terminates because of the nilpotence of G . It is

also the case that for each j with 1 < j < nil(G) -1, G.

nil(G) -1 is generated by U BA(i) . i - j We now give the definition of the Bruck loops .

DEFINITION 3.1.9: For i € N let (3^ = (x^,X0) and for

£ N let be the subloop of ft generated by ((3^: 1 < i < d)

We also define Bq = {(0,0)) *

Of course each Bd is a cexp-3 loop and d(B^) = d .

So we see that B^ = and Bg ^ , where we write = to abbreviate ”is isomorphic to.” 121

THEOREM 3-1.10 fBruck): For all d > 3, nil(Bd) - c(d) =

1 + [d/2 ] .

Proof: Given d > 3 let m(d) = 2[d/2] + l and

m ’(d) b 2[d/2] + 3. Given integers * • * j im , ^d) , not

necessarily distinct, with 1 < i < d for all j satisfying J ™ 1 < j < m'(d), we see by Proposition 3*1*7 that with

X = (p ,6. ,*.*,p. ) we have A (X) = h X2 V d ) [§]

(0,x X. X. ,,,X. - X. X. X. ...X. + X. X. X. ... X, ) 2 h ^ V d ) ll *3 H ln(d) \ H> il* ^(d) If d = 3 we choose i^, i2, and i^ to be distinct and if

d > if we choose i^ = i^ and i^, i^,i^, ..., j to be distinct

(which can always be done since m(d) -1 = 2[d/2] < d) ; we then

see that ( B ^ ^ / 2] ^ t(OjO)} hence, nil(Bd) > c(d) . On the other hand, with Y - (p. ,p. ,*.*,p. ) we have *1 22 V(d)

AP/d\(i) = (0,x X X ...* - X X X ...X + * ' 2 *3 H m'(d) X1 3 if m ’(d) x x x. ...x. ) . If i, , i ,.. .,i . , , are not distinct *1 *2 \ m* (d) 4 5 W then ^c^dj(Y) - (0^0) > otherwise since m'(d)-3 = 2[d/2] > d - 1 , at least two of the integers i^, i^, and i^ must be in the set

[i^,i5,...,im ^ dj) which forces A ^ dj(Y) = (0,0) also. Hence, by Theorem 2.2.10 and proposition 3*1*6, = t (0^0)) s-nd so nil(B,) = c(d) . □

We remark that the fact that (B^cfd) *= [ (0,0)) also follows from the more general result given in Proposition 2.2.17. Since nilfB^) = 2 we know that B3 * °81 ♦ We shall denote the set {1*2, ... , d} by Id .

------THEOREM 3.1.11: For ai1 d —> 4 a basis for b\ 3 is

[4/2] * E U U E where E «= [ (0, x. x, ,+ 3L ): 2 < j

and for 2 < t < [d/2], E* = {(0,xs ): S € P(ld) and js| = 2t}

[ dy/2 ] and hence |B^| = 3f^d^ where f(d] = f ^ 1) + £(2^J * t — 2

Proof: Given d > k we must show that every basic

t-associator for 1 < t < nil(Bd ) -1 = c(d) -1 = [d/2] is a

[

By Proposition 3.1.6, (P±,PjjPk ) = " xi,k + xi, j) and of course for this associator to be non-trivial (i.e. not equal to the identity) we must have i, j, and k be distinct. Note that E = ((px,p^,pk ): 2 < j < k < d] . If l

i + *1, _Xl,k + xl,p = tPl'Pk

(Px,Pj,P^) 1 * * Hence, every basic 1-associator is in the subloop of Ed generated by E. To verify that the elements of E are independent we write (p^, i V pk> - 123 d- i J ' (0, Z Z a[ (d,k),(j,,k')] xd_ji d + l-k’) and consider j 1 = 1 K — 1 * d ^ d the ( ” ) by (2) matrix whose entry in row (j,k) and

column (j',k') is a[(j,k),(j k ' )] (the rows are indexed

by { ) : 2 < J < k < d] and the columns are indexed by

((j',k'): 1 5: 3' < d-lj). Each row has only one non-zero entry

in the columns (o',k') where j* < d -1 (these are the first

(dg d) columns); and each row (j,k) has this non-zero entry

in column (d-j,d + l-k). So the rank of our matrix is (dg d) and

therefore the elements of E are independent. We give an example

for the case d = 5 :

* ^ 5 X 3,5 X 3 A X2,5 * 2 ^ * 2 , 3 X l,5 *1,3 x i,; (p1 ,plf,05 ) Z1 0 0 0 0 0 -1 1 0 0 (P1 , P 3 , P 5 ) 1 0 0 0 0 -1 0 1 0 (0 J ) I 0 0 1 0 0 0 0 -1 1 0

(Pl,P2,P5) \ 0 0 1 0 0 -1 0 0 1 i 0 v ° 0 0 0 1 0 0 -1 0 1 \ o 0 0 0 0 1 0 0 -1 1

Next, for 2 < t < [d/2] we consider the basic t-associator

T = ((...((PiJ0j,Pk),pje^ m )...)*0p*0q) . By Proposition 3.1*7

T = * * * xq - xi*kXi * * * Xq + XiXjXi ' * * Xq^ * In °rder that this associator be non-trivial, i, j, and k must be distinct. Furthermore, m, p, and q must also be distinct and |(i,j,k} fl (i,m, ...,q)| < 1 . If this intersection has cardinality 1 then w.l.o.g. we can say that i 6 ti,m,...,q) 1 2 k

and then T = ( O ^ x ^ x ^ ^ .., x^x^) € (Et> ; if the intersection

is empty then T = (0,x.x,x ... x„ - x.x, x ... x + x.x.x. ... x ) £ * 0 «: 4 q i x A q ijJt q y *\ (E^) also. By considering the former case we see that every * element of E, is a basic t-associator. Lastly we note that all [ d/2] * the elements of E U (J E, are independent. It is immediate t = 2 z , , . . _ [ d/2] that |E | = ( g 1 ) and |Et | = (£) , so d(B^) = (d’1) + Z (gd ) t = 2

d-1 [d/2] d ( p ) + ^ ( d) Thus [B^l = 3 2 t=2 2t _ □

pd -1 COROLLARY 3*1.12: |Bd | = 3

Proof: |Bd | = * |B^| . Now by Theorem 2.3.8

d(Bd/B£p = d > EO lBd^Bd I = ^ and lBd I “ ^ where

d -1 [d/2] d d -1 D = d + ( 2 ) + t ^ 2 (2t) = 1 + f*'1 ) + ( 2 ) +

C ^ ^ V l * + (Vt1) = r = £ d “ 1 ' □ t = 2 “ ’L t ’ =0 %

The following lemma is critical for what we shall do in the

remaining chapters of this work.

LEMMA 3■1* 13 ; If G is a c-^ exp-3 loop with basis

T = CYjl# Yg# • * • j Y^J then each element x of G has a unique ed e^ _ eg e, representation in the form x *= g’[y^ (y^ ~ (,..(vg y^ )...))] where g* € G ’ and 0 < e^ < 2 for 1 < i < d . 125

Proof: We consider the coset G'x . By Theorem 2.3.8

(G'Y-^, * • • ,G’vd) is a basis of G/G' and since G/G* is a 0 0 6 € group we have G'x = G' . [ Y{jd(Ydd_^1( • • * (Yg^Y^1) • * •))] and so ed ed 1 e2 ei in particular x = g'[Yd (Yd _ \ (- - -{Y2 Yx )...))] for some

g' € G r . Notice that the bracketing is very important when

ed ed -1 e2 ei we are back in G: for example, G'x = G'[vd (Yd „"i (*"(Y2 Y1_^___^

= G' t ( ( . . . ( v ^ d_-11)...)V22 )Yi1] tut 0 0 0 0 x = g"[ ((.-•CYddYdd-"^1)...)Y22 )Y11] with g" f g' in general. Now since |G | = |G1 | * |g/G* | = |G' J *3 , the representation of

x in the form S ’[Yd (Yd-1 ^***^Y2 Yi )**•))] is necessarily unique. □

FROFOSITION 3 * 1 » : Assume that d > ^ . If d is odd then

|Z(Bd)| = 3d and if d is even then [Z(Bd)| = 3 •

Proof: Since x0 * xT = (-1)P^S,T^XS UT for S fl T = 0, any element (b,y) of B^ certainly has the form (b,y) = d ( £ i+x, , £ kQx ) where j£+,k £ F for 1 < t < d t=i t t s ep(id) s " “ and S £ P(id ) ■ By Theorem 2.2.10, Z(B^) = t(b,y) € : for all i and j with 1 < i < j < d, ((b,y), (x^x^), (x^x^)) =

(0,0)) . So, (b,y) eZ(Bd) iff (b,y) € Bd and for all i and j with 1 < i < j < d, (0, £ k x x.x. + b(-x. + x.)) = s £p(id) **

(0,0) . We claim that if (b,y) € Z(Bd) then kg = 0 for all 126

S £ 'with 1 < |s| < d - 2 . Given such an S , choose

i and j so that {i, j} Pi S = 0 ; then we must have

ksxsu(i j} = 0 forcin£ ks = 0 * So' now have (b#y) e 2(Bd ) iff (b,y)€Bd , ks = 0 for all S € P(ld ) with l<|s|

and for all i and j with 1 < i < j < d,

k-jx.x. + 2 j£+x+(xi + -Ox«xh “ 0- Our l

all t 1 with 1 < t* < d (i.e. b = 0) * Given , since

d > i+- there exist i and j with 1 < i < j < d and

t' 4 ; and [krf - &. -£.]x x + £ a x (x. -x.) + a j i j i < t < d J t 4 ,xt, (xd - Xj ) = 0 forces = 0. Hence, (b,y) € Z(Bd ) iff

(b,y) e Bd, b = 0, ks = 0 for all S € P U d) with 1 < |S| < d - 2 ,

and for all i and j with 1 < i < j < dt k^x.x. = 0 iff — — ' (5 l j (b,y)€Bd , b = 0, and kg = 0 for all S £ P(I ) with

0 < |S | < d - 2 . Our third claim is that an element (c,z) € Bd

satisfies (c,z) € iff c = 0 . On one hand we have already

seen that (c,z) € B^ implies that c = 0. Assume that

(c,z) € Bd\Bd i by Lemma 3.I.I3, (c,z) = 0 0 0 0 S' * [Pdd * ( P ^ i 1 * (.-.(P22 * 311)-.-))] t o some g ’ 6 Bd and ed 0 for some i with 1 < i < d . Since multiplying

two elements of 6 involves only vector addition in the first

coordinate and since the first coordinate of g ’ is 0, we d _ see that c = Z e,x. = e.x. + Z e.x, £ 0 . Thus our t-l** 11 1< t < d t / 1

third claim is proved and we see that Z(B^) £ . Therefore,

by Theorem 3.1.11, Z(B^) is the subloop generated by Ej-dyg] (d ) ( d ) and |z(Bd)| = 3 2^-d^ . If d is odd then |z(Bd)| = 3 d“1 (d) 3d and if d is even then |Z(Bd) | = 3 d = 3 • □

The sequence of loops B^,Bg,Bg,... is indeed remarkable:

their cardinality becomes large, their nilpotence class becomes large, but their centers are always as small as possible, c-ty exp-3 loops having centers of cardinality 3 will be used in

Chapter VII to construct partially balanced incomplete block designs.

We now give a construction of another infinite sequence of subloops of ft .

DEFINITION 3«1*15: For each non-negative integer d the d subset K, of ft is defined by K , = {( Z X.x., Z — d i=l 1 1 S€P(ld) S S

€ F for all i £ 1^ and kg € F for all S € P(ld )J *

THEOREM 3»l»l6: Kd is a c-!?| exp-3 loop. Furthermore,

|K4|-3d + 2 and Bfl 5 K4 .

Proof: Since (0,0) € Kd , to show that Kd is a c-#f exp-3 loop it suffices to show that Kd is a quasigroup since 128

commutativity, exponent 3, and the Moufang property are all

inherited from (3. Consider elements (b,y) = d ' d { F i x , F k x ) and (b*,y’) «= ( £ 4'x , £ k ’x ) i = 1 S€P(ld) i =1 1 1 S €P(ld) S S d of K,: (b,y) * (b1 ,y') = ( F (X, + X ’)x, , F (k + k’)x + d i = l 1 1 1 SGP(ld) S 3 S d d ( Z (k,,-k')x )( z X.x.)( F f.'x )) 6 K, since for all SfP(ld) S S S i=l 1 1 i=l 1 1 d subsets S € P(ld) and ti,jj € (^^d^ satisfying S n (i,jj = 0,

S 1) {i,j) € P(ld) - We have seen in the proof of Theorem 3*1-5 that given (a,x),(c,z) € Kd the unique solution of (a,x) * (b,y)

(c,z) is given by (b,y) = (-a,-x) * (c,z) . Since (a,x) £ Kd implies that (-a,-x) € Kd we have (b,y) 6 Kd and therefore

d + 2d Kd is a quasigroup. It is immediate that ]Kd| = 3 and that bU , c^ K u . . □

j -j PROPOSITION 3.1.17 : |KJ[ | — 3 " " and d(Kd ) = 2d + 1 ; more precisely, a basis for the group Kd is ((0,xg):

S ^ P(ld ) and 2 < jS| < d] and a basis for Kd is

C(0,xg): S € P(ld ) and 0 < |S | < 1} U t (x^ 0): 1 < i < dj .

Proof: Let Sd = {(0,xg): S £ ^(ld) and 2 ^ Is | £ d) ^ d let Rd a l(0,xg): S € P(ld ) and 0 < |s| < 1} U

((xd,0): 1 < i < d) . Given S € P(ld) with|s| > 2 , choose distinct elements i,j € S and let S = S\{i,J} . We have

((0>x_)> (x±,0), (Xj,0)) = (0,x_xixJ) and ((0,x_), (Xj,0), (x±,0)) - s s s 129

(0,-x x x.) and one of these two assoclators is equal to S 3 (0,x ) = (0,x ) . Hence, K' contains the subloop SU{i,j) S d generated by §d . On the other hand the general associator

d d of Kd is (( E 4 ^ , £ ksxs ),( 2 T ksxs > > i =1 Sfp{id) i =i sep(id) d ( i x ' (x £ k"x )) = (o, £ £ - r . & " ) 1 =1 S 6P(ld ) 1 < i < j < d S €P(ld ) 3 3

k ' (Jt. - I + k" ( X . Z\ - I.£,1 )]x x. ) . So, K' is contained b l j J1 b l j J 1 & Ifd ^ in and hence Kd = ^ d ) * is immediate that the elements

of Sd are independent and so Sd is a basis of Kd * We have

d(K^) = 2d - (d + 1) = 2d - d - 1 and so |KJ | = 32 " d -1 . Now

d(Kd ) = d(Kd/ q ) = 2d + 1 since l^/K^J = 3d + ~ d ~ 1 =

2 cL 1 3 (recall that Kd/Kd is an elementary abelian 3“group).

The 2d + 1 elements of ftd are independent and hence are a

basis of K.d . □

Note that K = C_ and K, = . o 3 13

PROPOSITION 3.1.18: For all d > 2, Z(Kd ) is the subloop

generated by ^ - ((0,xs ): S 6 d - 1 < |s| < d) ,

and so |Z(Kd ) | = 3d + ^.

d Proof: Let z = ( Z , £ koxe) be 8111 arbitrary t -1 1 s e P ( i d ) s s element of Kd . By Theorem 2.2.10, z € Z(K_d) iff for all 130

i,J € Id, (zJCxi,0),(xJ,0)) = (z,(0,xi),C0,xJ )) =

(z, (xi,0),(0,xJ )) = ( z , ^ ) , ^ ) ) = (z, COJxi),(0,x^)) =

d (0,0) iff for all i j £ I,, 2 k^x x x = x.( E £+x )x . = a S €P(ld) Jt=ltt 1 , d _ d _ xrf( ^ = 0 ' First, E i^x^x.^ = 0 for all i f 1^ t = 1 t c= 1

forces i, = 0 for all t . Second, E L j u x .x . = 0 1 S €P(I^) 1 J

for all i, j 6 forces ks = 0 for all S € P(ld ) with

0 < [s[ < d - 2 since for any such S there exists a set

Ci,3) c Id\S . Hence, z € Z(Kd ) iff &t = 0 for all t with

1 < t < d and k = 0 for all s € P(Id ) with 0 < |S | < d - 2 .

That is, 2 (k ,) = ((0, £ .kC3xQ): k„ € F) and this subgroup a S €P(Id ) b b b d - i < |s [ < d

is generated by the independent set ^ . Finally |pd | = d + 1

implies that |Z(K, )| = 3^ + 1 • □

Note that by Propositions 3 .1.1*1- and 3.1.18 for aii d > k,

Bd D Z(Kd ) = z (Bd) • Also we see that iKg \ ~ 3^ and -

33 ; hence by Theorem 2 .3.19, Kg = Y^. = G8l ® = R(l,2) .

PROPOSITION 3.1*19: nil(Kd) = c(d) .

Proof: Since Bd c Kd we have nil(Kd ) > c(d) . And by

Theorem 2.2.10 it suffices to show that every basic c(d)-associator is equal to the identity element (0,0) . For each i with

1 < i < d let = (x^,0) and Y^ = (0,x^) and let Z » (0,x^) . 131

By Proposition 3*1*7 bhe only "basic c(d)-associators •which ore

not trivially equal to (0,0) are those of the form

((•••((x ^ x ^ y^ x ^ x^.-.) , xp>xq) *» ( ^ Y i Y A *** xpxq) and those of the form ((...((X^,X^,Z ),X^,Xm )... ),X ,X^) =

(0,x.x x x ...xx ) . The number of arguments of a c(d)- i « in ^ associator is 2c(d) +1=3+ 2[d/2] which is equal to 3 + d

if d is even and 2 + d if d is odd. In either case the

2c(d) elements x.,x..x., ... ,x can not all be distinct (since there are only d such elements) and hence all basic c(d)- associators are equal to (0,0) . □

Note that |k3 | = 311, [zfK^) | = |k^ | = 3^ (in fact

Z(K^) = since nilfK^) = 2 implies that £ Z(K^)) , and = ? * Now = 37 and is different from all the nilpotence class 2 loops Rfm^,!^) constructed in §3 of Chapter II. So, for any non-negative integer m^ — ^ Rfm^nt,,!^) = Rfm^n^) ® is a c-fy exp-3 loop of nilpotence _ llm_ + J+-HL. + nip class 2 satisfying ^(m^m^m^) | = 3 ,

+ dl _

In particular consider R(l,0,3) and R(8,5,0): |r (1,0,3)| =

iR(8,5,0)| = 337 and |z(R(l,0,3)) | = |Z(R(8,5,0)) | = 313 , but ^ 1 R(l,0,3) and R(8,5,0) are not isomorphic since |R (1,0,3) | -

313 and |r '(8,5,0)I = 38 * Also |R(0,5,1)| = |R(^,0,0)| = 3l6 and |r ’(0,5,1)| = |r ’(U,0,0)| = 3^ , but R(0,5,l) and R(^,0,0) 132

are not isomorphic since [z(R(0,5,l))| = 3^ and |z (r (4,o ,0 ) )| = k 3 * There are infinitely many examples of these types.

§2. The Maibos Variation

In this section we give a very brief discussion of the

construction used by Malbos in [ 4l] to prove Theorem 2.3,22.

DEFINITION 3.2.1; Recalling Definition 3*1*1* we define CO # f the odd part of 5 by 0(5) = & A ^ + ^ = { £ k_xr, : j£ € N, n , n d i = 0 n =1 n

k € F. and |S I is odd for all n with 1 < n < Xj • n * i n 1 — —

Although Malbos approached his construction and proof from

the point of view of medial spaces as studied by Soublin in [57],

we give here the outline of a loop theoretic proof. A key result

used constantly in the proof is the following.

PROPOSITION 3*2.2: 0(5) is also a vector space over F on

which the multiplication of 5 restricts to an anti-commutative

operation; that is, ab = -ba for all a,b €0(5) (of course ab

need not be an element of 0(5)).

Proof: It is immediate that 0(5) is a vector space over

F . If S and t are disjoint sets of odd cardinality then p(S,T) p p(T,S) modulo 2 implies that xg • xT = -x^, • Xg . The 133

anti-commutativity for 0(5) follows from the distributive

laws. D

THEOREM 3.2-3 (Malbos): M = (0(5) x 0(5),=) where the

operation ° is defined by (a,x) ° (b,y) =

(a + b + aby - abx , x + y + axy -bxy) is a exp-3 loop.

Proof (outline): The anti-commutativity of 0(5) gives

M its commutativity. (0,0) is the identity of M, the inverse

of (a,x) is (-a,-x), and (a,x) o (a,x) = (-a,-x) . Since the

disjoint union of three sets of odd cardinality also has odd

cardinality, 0(5) x 0(5) is closed under the operation « .

(a,x) * (b,y) = (c,z) iff (b,y) = (-a,-x) <» (c,z) , so M is

a quasigroup. A lengthy computation shows that

(a, x) - [ (a,x) o ( (b,y) ■> (c,z)] ] = (-a + b + c- a(b + c)(x + y + z)- bc(y - z) - abc(xy - xz -yz)^-x + y + z + (a + b + c)(x)(y + z) + (b - c)yz -

(ab - ac - be)(xyz)) = [ (a,x) . (b,y)] • [ (a,x) • (c,z)] } so M is indeed an 27< loop. C

PROPOSITION 3 .2 (Malbos): The associator in M

( (a,x), (b,y), (c,z ) ) = (-abz + acy - bcx^ ayz - bxz + cxy ) and

(M'jM'jM*) f {(0,0 )} so M ' is not a group.

Proof (outline): The value of the associator is verified by a direct computation; and by taking the eighteen distinct elements x^,x^,...,x^g of and letting ^or 13J+

1 < i < 9 one computes that ( ( X - ^ X ^ ) , (X^X^Xg), (X ^ X ^ ) ) /

(0,0) . □

In the proof of Theorem 2*3*22 Malbos takes the 2d elements

and defines to be the subloop of M

generated by ((x^x^): 1 < i < d} . It should be of great

interest to analyze the loops more closely; in particular,

how close is to being the free c-/7? exp-3 loop on d generators?

It is interesting that Marshall Hall, without knowing of Malbos'

work, very recently in a private communication proposed this

same question for the loops Hd which are the subloops of B

generated by {(x^x^): 1 < i < d) , Also there are analogues

in M of the subloops of (3, and these should be interesting

as well. Although the question has not yet been settled, it is likely that the Triple Argument Hypothesis is valid in M . CHAPTER IV

COMMUTATIVE MOUFANG EXPONENT 3 LOOPS

OF NILPOTENCE CLASS 2

§ 1. The Behavior of the Product

We have already seen infinitely many examples of

exp-3 loops of nilpotence class 2, namely the loops Rfm^mgjin^)

discussed in the final paragraph of §1 of Chapter III. In this

section we shall completely determine the product rule which any

finitely generated c-JT? exp-3 loop of nilpotence class 2, G,

must obey. We do this in terms of the unique representation

of the elements of G in the "left to right outward in"

bracketed form of Lemma 3*1*13 with respect to the ordered

basis ? = (Vd,Vd . 1i-**jY2;V1 ) (that is, r = (Yd, Yd _ ^ .. .,y2, y±)

is a basis of G ) . In fact we can identify y = e . e p e Sy[y^ 9 (- * • ( Yi^ )...)] with the ordered (d + 1)-tuple

e . e e ( < ^ which is an element of

G ' x . + 1=1

DEFINITION ^.1.1: We shall be concerned with the multi- e 2 Cl j e j o e Z 1t plication of two elements of G, z = Sz[ Yd * (- - - Cv2 9 Y-^ 9 )•••)] e e e and y = g^I y ^ ’ (...()•••)] where by Lemma 3.1.13

135 shall write E(z) = (eZjd'ez,d - 1'' * *■»ez, 1^ and E (y> =

(e^. (j#ey d - l J ' " ,ey 1^ * We de;Cirie f°r 1 ^ < b ’ < t" < d

AZtyft *jt) = ez,t'ey,t " ez tey t* Wlth the

and subtraction performed in GF{3) and ft! (t",t',t) = ~ 2- » y ______(e , „ - e (t',t) . We shall often be writing z = x. z,t y,t z^y i and y = x . in which case we shall abbreviate e . by J

ek>t and E ^ ) by ER for k G (i,JJ ; ^ ^ by ;

and %L.x. v . by 1?L , - When there can be no ambiguity we x i> xjJ JLuL shall even suppress the subscripts and write just A(t',t)

and Wi( t",t',t) . We shall come to think of ,t) as

a "Moufang function" in the sense that it formally satisfies

the ^identity.

DEFINITION V.1.2: Given y and z as in Definition l+.l.l, we define 7 ^ = y*> " X(... (V g ^ Y ^ 1 ) • - ) ) and 0 0 0 0

ffytt ~ Vty,t( J ~ 1(. * • (y ^ ^ Y ^ 1 ) • * •)) for 1 < t < d .

♦ We also define f (k) = e , + y , and F (t) = z,yv J z,k z,k z,y 1 f (t) f (t -1) f (2) f (1) Yt ^ (YtZ-'I (***(V2Z'y YlZ,y in the case that z = x. and y = x. we write TT. . for TT_ . , TT. . 1 J lj V Z y w J f t LEMMA, lj-,1.3: If g is a c-ty loop with nil(G) = 2 then

there is simplified expansion (see the paragraph following

Theorem 2.2,7) for aH associators of G.

Proof: Given w;x,y,z 6 G, "by Theorem 2,2,7

(wx,y,z) = (w,y,z)(xJy,z)[ ((w,y,z),w,x)((x,y,z)Jx,w)] . Since

G2 = C (w#yjz ),wjX) = ((x,y,z),x;w) = e implying that

(wx,y,z) = (w,y,z)(x,y,z) . D

Now the proof of the following result is immediate.

COROLLARY 4.1 A : If G is a c-77? exp-3 loop with nil (G) = 2 and if > * * * > ^2^ * * ^2 * * * * ^t11

t t' t" are elements of G and if TT x. , TT y. . and TT z, are 1 = 1 1 3= 1J k = l* any products (regardless of "bracketing) of the elements

, y ^ .. .,yt , , and z ^ - . ^ z „ respectively then

t t' t" t t' t,T ( TT x , TT y , TT z ) = TT TT TT (x ,y ,z ) . i =1 j =1 J k =1 i =1 j = 1 k = 1 J

LEMMA 4.1.5: If G is a c-5^ exp-3 loop with nil(G) = 2 and

^ = g2[Yd2,d(...(Y22j2V12,1)--0] and ^ =

6 0 6 g^[ * * * (y21,2^11* 811:6 elemen‘tB of G "then f°r aU- t 6 6 with 3

Proof: Note that since G' is a group it is not necessary to bracket products of associators. First we see that

^Yt 9 ,Tb,t-i,Ti,t^ = ^Yt 9 *^ t -i,TTi,t-i^Yt 9 '^t-i,Yt 9 ^

(Yt2,t^ Jt - 1'«Lft - l )(V 1t, t - 1*Yt>°2't + ei,t =

^Yt 9 -i> \,t -i^ ' So'

(yt * jT^, t - i ,Tri, t ^ Yt 9 jTTi,t -1'-i^ = 6 “6 (Yt 7 'Tfe,t-i,Tri, t - i ^ Yt 1 ,T^,t-i'ni,t-i^ = 6 - 0 (yt2ft t ~ i ,T!i t - ' Wow by Corol±a^y eo 4- - eT +■ t ■ 1 t -1 e 4. - e + eo „ e

l

TT ( Y ^ t " ei't,Y’ei'P,Yq2,q) = l

TT (yt2^ ei,t,Yi2,1,Yj1,J)(Y ^ " ^ Y ^ ' S y ^ ) « l

TT (y+*Y.;V.) ' * Recall that throughout these -1 * 1 J computationsj particularly regarding the penultimate equality, we use the properties of associators given in Theorem 2.2.1. □ 139

DEFINITION U.I.6; If G is a c—^ exp-3 loop of nilpotence

class 2 with ordered basis f = ( V Yd - 1," ,,Vi) then ve define _ d X(G, r) to be the set G ' x X ( C v rl .,} > endowed with the i =1 +±“1 0 0 6 binary operation * defined by Vd2,d, Yd2_'d ~ \ Y ^ ' 1 ) *

(o v6l,d el,d-l el,l. , , . fd fd - l fl. (SjjVd t Yd _ 2_ ^***,Y1 ~ 2* l^^d * Yd — 1 * * * * * Y1

where v (E2,E1 ) = Tt (y^ V ^ v J ' 1< j

THEOKEM 4.1.7 ; If G is a c-ty exp-3 loop of nilpotence

class 2 with ordered basis T = (Yd, Yd _ ^ ’ *' * Yi) ^hen G is

isomorphic to x(G, T) *

Proof: It suffices to show that the bijection

*: a h. c x J 1 eH eH i ep ei -| eQ en g[Yd ^Yd - 1 yi )**•))] (g,Yd >Yd_i ,y± )

given by Lemma 3.1.13 is a homomorphism. That is, we must show

that given £ G with Xg =

g2[Yd2jd(Yd2-l ” 1(**'(Y22,2Yl2'l5' " ))] and X1 =

r 9l,d/ ei,d - , el,2 ®l,lx gltYd ^Yd. -1 C - - - (V2 Yj. )•••))] we have Xg . x_L =

(g2elv(E2-'El))Fd * We make the following claim: for all k with

% -,(t,i,d) 3 ^ k ^ d^ k ,TTik±,k = ( i < j < i n < t < k J >Fk* K

We establish the claim by induction. First we consider the case € 6 € 6 (y3 ,3>1fe#2,Ti, 3 ^ Y3 ,3,Ti,2,Tfe,2^Y3 ,3^Y3 ,3^ Tb,2Ti,2-'^ since 6 6 nil(G) = 2 implies that G' £ Z(G) . Now Y32,3(y31,3[ gfl^ 2 ]) =

f f f f x ^ 2 1 Y3 2^L 2^ = Y3 ^Y2 Y1 ) " F3 > the first two e^uaJ-ities following from the diassociativity of G ; and by Lemma 4.1.5

®p p q tyn -i(3jijd) (y3 ' ^2,2-’TTl, 3 ^ Y3 * = K j

»fe x(3,2,l) CY3,Y2,Yl) j • Hence, Tt2 3 • Tt^ =

C TT (Y-ujY-jY-j) 1 )F~ . Next we consider the 1 < J < i < t < 3 1 value k with 3 < k < d and assume that the claim has been e2 v established for the value k -1 . ^ ^ = ^Yk * ^ k - l ^ l j k llfl

by the diassociativity of G and since G' c Z(G) * Now, by

© € Lemma if. 1-5, (vk * , T ^ k _ ±, TT1^k )(vk ' '\ k - 1'^ , k - 1^ =

TT (y^-jVjjVj) * - Anci by induction l -1 1 < j < i < t < k - 1 1

Hence, since G' £ Z(G) and G is diassociative TTg k * ^1 k ~

( TT ( Y ^ Y ^ y J j1 1 < j < i

TT ( Y ^ Y ^ V J ,X )tYk\ ,] = l

( TT (Yj-jY-^Y-) * )F. . So the claim has been l

established. Now, ^ • x± = (figT^ d ^ l ^ d ^ = d^l, d^ since G' c Z(G) . Hence, by our claim with k = d we have that

% ,(t,i,j) Xp *x = (gpg. )[( TT (Y+, Y.:, YJ f )Fj ] = 1 ^ 1 l

(g2glV(E2 ,E1))Fd again since G f = Z(G) ; this completes the proof of the theorem. □

We remark that what Theorem U.l.7 tells us is that the product rule for G is very nearly a direct product rule, the only difference being a "twist" in the G*-coordinate which is completely determined by the (d) basic associators (y^,y^Yj) with l

mention this again in Chapter V. We have stated Theorem 4.1.7

for nil(G) = 2 , but of course it is valid, for nil(G) < 2 :

when nil(G) = 1 the G 1-coordinate is vacuous and the product

rule is the standard direct product rule for elementary abelian

3-groups.

In light of Theorem 2.1.1 it would be interesting to

discover whether Theorem 4.1.7 has an analogue for c3-loops

in general. Lemma 3.1.13 depends on part (ii) of Theorem 2 .3 .6 ;

so to obtain such an analogue one would have to work with the

Frattini subloop rather than the associator subloop.

COROLLARY 4.1.8: If G is a c-#f exp-3 loop of nilpotence class 2 with ordered basis r = CYd,Yd^ Y ^ ) then the associator (x ,xQ,xn ) = TT (y,, y., Y * ^ where J ^1 l

A(t,i, j ) = e^ t^3 2 ^*^ ^ ** t^3 ®3 t^2 *

Proof: By Theorem 4.1-7 we can work in the isomorphic loop

6 +6 6 + e X(G,?) . Xg *x1 = (g2g1v(E(x2 ),E(x1 ))Jvd2#d 1,d, 1,J

= (g3g2g1v(E(x2 ),E(x1 ))v(E(x3),E{x2 *x1 )) ,

e„ . + e + e e + e + e 3,d 2 ,d i,d 3,1 2,1 1 ,1 . . Yd >*"'Y1 * *

x3 **2 = (G3£2v(E (x3 )>E (x2 ))>Yd3,d S,d,...,v13,;L 2 , ± ) ‘> and Ib3

< V * 2 > * * i = ^g3S2giv^E^X3 ^ Efx2 ^ v^Efx3 *x2)^e(xi)) t e0 . + e0 + e e_ , + e_ + e, .. Vd3' ’ ^ ,...,v13>1 jl 1’1 ) . Since G'CZ(G),

A f t/ i "j ^ (x^x^x^) = TT ^t'^i^j) * is the uniclue element

of G1 satisfying vfEfx^Etx^)v(E(x3),E(*2 *x1) Kx^x^,^) =

v(E(x3),E(x2 ))v(E(x3 *X2 )^E(x1 )) . That is,

K If .(Vt.VyVj)

\?,X2 *x J

t 1 J l

* x2,x1 (W V Taking for all values

of t, i, and A(t,i,J) = + *x2, x ^ t'i>'J) "

^ ^ X i (t,i,o) - “ ^e3 ,t " e2 , t ^ 3,2 ^i# +

fe3#t + e2,t ‘ el,tJ[(e3,i + e2,i)el,j * (e3,0 + e2,d)el,i] "

(e2,t - el,t^2,l^i,li^ " ^e3,t “ e2,t _ el,t^e3,i^e2,j + el,P e3 , ^ e2,i + ei,iJ] = fe3,t " e2,t^ ■ fe3,t " e2,t ' el,t)]

A3,^)t (03,t + e2,t ~ % t ^ - ^e3,t - e2,t " el,t^ +

A2,1(i,'3)[ (e3,t + e2,t “ el,t^ " (e2,t " el,t^ =

BXfth3fZ^±f^ ~ e2,tA3,l^i,J^ + e3,t^,l^1,;*^ filveS thC des±red representation for (x^XgjX^) • □ Ikk

Compare the form of the associator in general c-5?? exp-3 loops

of nilpotence class 2 given in the above corollary with the forms

of the associator in the Bruck and Malbos constructions given in

Propositions 3*1*6 and 3*2.4 *

§2. The Free Construction

We shall use our knowledge from § 1 of this chapter about

the structure of c-7% exp-3 loops of nilpotence class 2 to

construct many such loops. Heuristically speaking, we shall

show that a set of (d + l)-tuples provided with the product

rule of Definition 4.1.6 is in fact a c-7?l exp-3 loop of

nilpotence class 2 and dimension d .

DEFINITION *1.2.1: Given d > 3, we let denote the

09 infinite elementary abelian 3 -group & c_ = Cq ® C„ <8 C_ ® ..»j i =1 J J J ^ ^ ^ for each i with 1 < i < d we let Gi « [y^ Y ^ Y ^ be the cyclic

group of order 3 generated by y^, written multiplicatively; that 0 0 f 0 0 t is, y^ * - Yi i and since no ambiguity will arise we shall

denote v° by e for each i with 1 < i < d and also vf i - - - i * ^ by y^ i and we define = A x ^ ^ +1 i* ^at

6 6 € Td = C(a,Yddj • * -Ai1): a f A* and 0 < ei < 2 for all i with 1 < i < d) . The identity element of A will be denoted l k ‘j

-H» — by e . For each a £ A we let a denote the element of — a — * — -ft given by a = (a,e,e, ...,e) and let A = (a: a

k € (0,1,2} we let

k / * k^ Yx = (e ,e,e,...,e,e,v1 )

k / * k v YP = (e ,e,e,...,e,y ,e)

k k. y, = (e , Yd>e> *>e>eje) and for aH i with 1 < i < d we Td>

let CL = (y^ Y ^ Y ? ) •

Please note that we have no loop at hand yet and that in the following definition (y^Y^Y^) is an ordered triple, not an associator.

DEFINITION b.2.2: Let B = { ( Yj_ , Y.. Y.): l<.i

(Y.».»Y.,»Y.>)V by [ Yo-» Y-»Y_.] which will be called the associator x l J x -i j ------symbol of y^, Y^> atld yj • shall not be concerned with * elements of A which are not in the subgroup generated by

, so we define A = , A = (a; a 6 A} , and d S = A x X G. . .

We shall define a product * on the set in such a way that (Sd**) is a c-% exp-3 loop of nilpotence class 2. We shall 11*6

see that the (d) associator symbols will correspond to the

(d) basic associators which are the generators of the associator

subloop of (Sd, *)j and hencet A will be the associator subloop

of (S^, *) and tV Yd - r " * ^ i ) will be a basis of (S^,*) . The remarkable aspect of this result is that we obtain a loop

with the desired properties (and a corresponding HTS) regardless

of how the values [Y+,Y-.>Y-] of the associator symbols are t i j "assigned" by V . The sole requirement is made merely to

insure that the loop (Sd,*) is not a group. Furthermore, it is by "assigning" values to these symbols in an appropriate way that we gain control of some properties of the loop we construct, for example the size of its center. Note that the operation * will depend intimately on V , so that strictly speaking we should write *

Theorem U.l-7 tells us how the operation * should be defined.

DEFINITION **.2.3; The binary operation * is defined on

er - - / 02.d e2,d -1 e2.1\j 1. Sd as follows: given Xg = (»2,Yd * ,Yd _,1 ' ) and

, el,d el.d-l 6l.lx *i ■ <*l'v4

We remark that since for all t, i, and j with

1 < j < i < t < d, [Yt,V1,Y;j] € A , certainlyh .(E2,E1) 6 A and * is a well defined binary operation on . We make immediate note of the following facts.

PROPOSITION k . 2 A : (A,*)=A; for all r with l

(Gr, *) = Gr i and for all r and s with 1 < s < r < d v(G r x G . *) y = G r « G s .

Proof: If = (a2,e,e;...,e) and = (a ,e,e,...,e) then e2 ^ ^ = 0 for all k £ I^ ; hence 57^ ^ft^ijj) = 0 for all values of t, i, and j which implies that p,(Eg,E^) = e . So, x2 * xt_ ~ (a2aiJe^eJ * * ‘>e) anc* ^ijection -jf e 2 i* \|r^: A -*■ A is an isomorphism. If Xg = (e ,e, .. .,e,Yr f ,e,...,e) a *> a and — (© ,e, .. .,e,Yr * then ep ^ — e^ ^ = 0 for all k € Id\{r} and again j^ft^i,,]) =

(e2,t - % t )(e2,iel,J - S2,del,i) _ ° fOT a U ValUeB °f * t, i, and j implying that ^(Ep,E^) = e . Hence Xg *x^ * fr (e ,e, ...,e,Yr ,e,...,e) and the bisection G + Gr is

Yrr ~ Yr r an isomorphism. Similarly, in the case that Xg =

. * e2,r S2,s . . (e , e, ...,e,Y^. and x^ r e, e (e ,e,...,e, y f ,e,...,e, Y„ J je,>***#e) w® ""*«have -2,keg ^ ”*- '"l,ke_ - “ 0 for all k € Id\(r,s) and so 1(t,i,j) = 0 for all values of 148

t, 1, and j . Hence, = e*, *x_L = „ f f / r s ^ {e ,e, ...je;y ,ej •••,&), and the bisection r s Gr x Gg •+ X Gg is an isomorphism. □

(V^Y^) - ( Y ^ )

The last part of the preceding proposition is part of the

diassociativity of (S^, *) and can not be extended to products

of more than two of the • The following major theorem is

proved by the nine lemmas which succeed it. It is essentially

the converse of Theorem 4.1.7 • Throughout the proof we write

^ for k 6 u,a’3) ■

THEOREM 4.2.^ : (S^,*) is a c-!7( exp-3 loop of nilpotence

class 2 with basis {Yd, Vd _ ^ - • -, in which (y^Y^Yj) =

0 0 0 0 [Y^Y^YjJ and 5 * [Yad * (Ydd.‘i1 *(-"(Y2a *Y11)...))] =

( ed ed-l e2 el. (a7 Y(j * Yd -1 3 * * * 3 Y2 3 Y1 ' *

LEMMA 4.2.6; (S^, *) is commutative.

Proof: Since for all i £ I., e. . + e0 . = e_ . + e d* 1,x 2,i 2, i l,i f f f d d X 1 *1**2 = (aia2 ^ El,E2 ^ Yd 3 Yd -1 >***>Yi ) * So> 11 sufflces to show that h(E^,E2 ) = p,(E2,E^) since A is abelian; that is, it suffices to show that for all values of t, i, and j

\ f2 ^ 3i3^^ = ^i^>^J) * Now ^gC't#1^) = 149

(el,t "e2 ,t)(el,ie2 ,j _ei,je2 ,i) = (e2 ft - ei,t^e2 ,iei,3 - e2 f3eif±') . □

LEMMA 4.2.7 : (S^,*) is a quasigroup.

Proof: By Lemma 4.2.6 it suffices to show that given

6 Sd there is a unique x^ 6 satisfying *x^^ = x^ .

Necessarily, is uniquely determined; that is, for each

i € I .> e_ , is uniquely determined by ep . + e . = e . U 1 e:,ll, 1 3ji

(an equation in GF(3))* Then with i, j ) b = a? TT [ Y+> Yj j "Yj] 9 > since A is an l

LEMMA 4.2.8: (S^,*) is a loop with identity element ^ ■X’ e = (e ,e,...,e) in which the inverse of the element e , e -e, -c, / . O. 0 \ . , , "1 / *1 u 1 > X = (a»Yd ,) 1S given by x = (a , yd , ) .

Proof: Since — (t, i,j) = 0 for all t, i, and j , we x, e have u(E(x)^E(e )) = e i'or all x 6 and hence, x * e = x for all x 6 . For all values of t, i, and jj j_(t,i, J) = x,x~ [et - (e±)(-ej) " (ej)(-ei)] - 0 . Hence ^(Efx^Etx-1) ) = e* which implies that x *x”^ = (e*,yd,...,y°) = (e*,e, «..,e) = e* . 150

-Jf And since (S^,*) is commutative e * x = x and

x "*■ *x = e* for all x £ S , . □ d

LEMMA If.S.9: (S^,*) has exponent 3 ■

Proof: Since is comutative we need only verify

that (x *x) *x = e for all x = ) £ Sd .

^ x (tJi,j) = (et - et )(eiej “ ©jei) = O for all values of t, i, and j . Hence, n(E(x),E(x)) = e implying that 20 20 “6 —0 x *x = (a , Yd t • • • j ) = (a *\i ) — x j and by Lemma 4.2.8 the proof is complete. □

LEMMA 4.2.10; (Sd7*) an ^ 1°°P*

Proof: We verify the ^-identity: x3 * [x3 * (Xg * xx )] = (x3 * Xg) * (x3 * x1 ) . x2 ^x1 = (a2a1n(E(x2 ),E(x1 )),yd2'd X>d, . ..,y ^ ' 1 + , x3 *(x2 *x1 ) = (a3a2a1fi(E(x2),E(x1 ))u(E(x3 ),E(x2 *xx )) ,

j H- 6a j 6_ j 6^ _ “f- 6a _ + G-. Yd 2'd i • • • r 2 A 1 A ),

X3 * ^*3 * ^*2 * * 1 ^ = ^a3 a2ai*1I/ \l * * *»•••>

YjL 3,1 2,1 1'1), where ^ = n(E(x2 ),E(x1))(*(E(x3),E(x2 *3^))

^(E(x3 ),E(x3 * (x^ * x 1 ))) ; x3 **2 = (a5«2^(E(x3)#E (3,2))#Yd ' * * ) t 151

x3 = (a3a1n(E(x3),E(x1 )),ydJ'ue3,d + el,d^ ^ *3,1 * + *1,1! L) ,

(*3 *Xg) *(x3 *Xi) = (a"1a2alMRJyd 3'd 2'd ljd,e..,

—6 +6+6 Y2 3'l 2,1 1'1 ) , Where ^ « ji(E(x3),E(x2 ))ji(E(x3),E(x1))

ji(E(x3 ^x ^ E f x ^ *x-^)) ■ Hence, it suffices to show that

Mr = Mtj • V* = TT [Y+JY.,Y,] and l

%(t,i,d) Mp = TT [Y^Y^Y.] where 1

\ ,Xg +

\ 3,*3 *Cx2 *x1)(t*i' ^ " d +

+ *>(x *x2 ) *(x3 ■ Hence, it suffices to show that for all t, i, and j with 1 < j < i < t < d

?^(t,i,j) =/7^(t,i,J) . Recall that these are equalities in

GF(3) * Now,

J) = (e3jt ~e2, t ^ e3,ie2, j -e3,Je2,i^

+ (e3,t “el,t)(e3,iel,J “ e3,jel,i)

+ ^ e3,t + e2,t^ - ^*3,t + el,t^

C(e3,i + 62,i)(e3,j + el, ‘ Ce3,d +02,d)(e3,i + el,i)]

= (e3,t *e2,t)(e3,ie2,j ~ e3,AeZr^

+ f*3,t -el,t)(e3,iel,j “ e3,jal,i)

+ (e2 , f el,tK e 3,iel,j‘ e3,.Jel,i + e2,ie3,d ’

e2JJe3,i + 02,iel,j "e2,jel,i> 152

= (e3,ie2,J - e3,de2,i)[ (e3,t ‘e2,t> - ]

+ fe3,iel,j -e3 , j % i ^ fe3,t -el,tJ + (e2,t "el,tJ1

+ (e2,t "el,t)(e2,iel,j " e2,del,i)

= (e2,t “el,t}(e2,iel,j “ e2,jel,i>

+ fe3,t + e2,t + el, t ^ e3,i^e2,J + el,J^

On the other hand,

\(t,i,j) = (e2,t “el , t ^ e2,iel,j

+ ^e3,t - ]te3,±(e2,d +C1,J}

“ e3,j(e2,i + el,l)]

+ [e3,t ~ fe3,t + e2,t + el,t^[e3,ife3, j +e2, j +61, j5

' e3,j(e3,i + e2,i + el,i)]

= fe2,t _ei , t ^ e2,iei,j "®2,3*1,1^

+

- e3,j(02,i + el,i)] '

Hence, ^(t,i,j) = ^(t, i,j) and the proof is complete. □

LEMMA h . 2 . H ; For all t, i, and j with l

(v v\j *= t v W *

Proof: We shall show that [y. *(y. *y.)] * [ Y+, Y-.» Y-] = t i 0 t l j (Y^. * Y^) * Yj • We have and [V^V^Yj] = ([ Ytj • By Proposition b.2.b

— — * Y^ * Yj = (6 ...,e;Y Y^jej * * • je) and

— — * Y^ * Y^ = (e #e* • • • ,»e j ^6 j . ..,e,Y£jej***jS,e,e, ..«,e) .

Next n(E(Yt ),E(Y, *YJ) = e* since (t,i,j) = J V Yi *Yj

(l-0)(0 *1 - 0 *1) = 0 and n(E(Yt *\),E(Vj)) =[Y^Y^Yj] since (t,i,j) = (1 - 0)(1 • 1 - 0 • 0) = 1 . So, Vt* V Yj

Yt *(y± *Yj) = (e ,e, .».,e,Y^6j'*»je,Y^>e, ...,e,Yj,e,...,e) and

(Y^. * Y^) * Yj = ([ Y^^ Y^j Yj 31 ej * * * je> Y * * *t ej Y^^ej * * • t ef Yj je*

= [ Y^. * (y^ * Yj )] * [Yt,Yi,Yj] since

fi(E(Yt * (Y± *Yj )),E([ Yt,Yi,Yj] )) = e*. □

Note that Lemma b .2.11 shows that p, = v where v is the function of Definition U.1,6 for the loop x(G, T) where

G = (®

LEMMA k.2.12: nil( (Sd, *)) > 2 . Proof: By definition there exist t, i, and j with , * _ tvt^Vi,Vj] t e ; and so by Lemma 4.2.11, (y^Y-^Yj) f e .

Hence, (S^, *) is not associative. Q

LEMMA 4.2.13: { ^ ...,y^} is a basis of (Sd,*) and nil((Sd, *)) < 2 .

Proof: By definition of the operation *, tYd>Vd generates (Sd,*) and furthermore, for all i € Id

Yi ^ > • Hence, (yd,...,v1) is a basis of

(Sd, *) . By Lemma 4.2.11, (Y^Yj^Yj) 6 A forl

Hence, we can complete the proof of this lemma by showing that

A £ Z((Sd, *)) . By Theorem 2.2.10 it suffices to show that for all i and j with 1 < j < i < d and for all a £ A / — , ~ (a,y.y\.) = e . We have ^ J a ® (a,e,...,e,e,e,...,e,e,e,.,e)

Y^ ~ (© } 6) • • • ter e> * • * > ®®t e> * * • ) ® )

Yj ~ (e ,e,.. .,e,e,e, •. »,e,Yj* • • / ®) •

— _ -Jt By Proposition 4.2.4 and the fact that n(E(a),E(yd)) = e ,

Yd *Yj = (e >e>***jejYdjej***je>Yjjej**»je) and

a * y^ = • * ■; Y^;®> ••*,6,6,6,.i.,e) *

So, we see that p,(E(a ^Yj J^Yj ) * *7^)) * e* and 155

which is to say that (a^y^y^) = e . Thus (Sd, *)2 = {e }

and nil((Sd,*)) < 2 . □

LEMMA 4.2.14: For all a € A, for all k € Id , and for

all e^,e2, . . . , w i t h 0 < e.^ < 2 for all i € I ,

ek ek - 1 e2 el a#tVk * (Vk_! * (■••(V2 * Yx )-..))] =

ek ak _ e* * * * > e> Yk t Yk _ • i ) •

Proof: The proof is by induction on k . For each k € I d 6 6 6 6 let Wk = a ^[ykk * (ykk_-1 *C-.(Y22 * By

* Proposition U.2.U and the fact that n(E(a),E(x)) = e **or ali

x € , the conclusion is valid for k 6 tl,2) . Now, consider

k with 3 < k < d and assume that the conclusion is valid for

k-1. Since (S^,*) is conmutative and a £Z((Sd, *)),

Wk - (a * \ k )*[VkV 11 * (..•(Y2a *V11)— )] ■= Ykk *Wk . 1 .

So, by induction Wk = (e *, e,...,e, yk ek ,e,...,e)*

ek - 1 Gk - 2 0 l (a^Y k — d *Yk _ 2 * * *•*Yd ) ~

0k ek ek 1 ei (a^(E(Yk ),E(Wk _ 1 )),e,...,e,yk , Yk _ ^ Y2 ) • But

^ (t,i,j) = 0 for all values of t, i, and j: note

0k „ Yk ,Wk - l

that in particular, __ (k,i, j) = (1-0) • (0 • 1 - 0 • 1) » ek \ - 1 for all i and j with l

ek * ti(E(y^ ),E(Wk )) = e and the proof is complete. Note that

6d el in particular when k = d, we have Wd = (a, yd ) which

is part of the statement of Theorem 4.2.5 . □

In proving Theorem 4 .2.5 I have used notation involving the

1s to stress the relationship between Theorem 4.2.5 and

Theorem 4.1.7 * However, I have also used the notation

^{Efxg ),E(x1 )) rather than to indicate that what the construction depends on is the exponents. In fact, Theorem 4.2.5 could have been proved in exactly the same way by defining

T^ = A x GF(3) J = (s ,0,0, .. .,0,1,0 , .. .,0) j T i place from right to left ■Jf A arbitrary values [ y^, YjyYj] 6 A ; Sd = A x GF(3) where A is the subgroup of A generated by t[ YtjYijYj]:l

We remark here that in Zassenhaus* original construction of

Gg1 in [4] (see [34] for a description in English) he mentioned that although his construction of a finitely generated

"using" any finite commutative associative ring of characteristic

3 (it would still be much less general than Theorem 4.2.5) .

However, Theorem 4.1.7 shows that every finitely generated c-JPf 157

exp-3 loop of nilpotence class 2 can be constructed by the method

of Theorem 4.2.5, ’which "uses" only GF(3) in the sense that

the operations of multiplication and addition involving the

exponents are the operations of GF(3) * Hence, the more general

rings need not be considered.

We shall often use the phrase, "every finitely generated

c-ty exp-3 loop of nilpotence class 2 can be constructed using the method of Theorem 4.2.5," so we shall explain this precisely here. If G is a c-771 exP~3 loop of nilpotence class 2 with ordered basis r >= (Yd,. ..,y^) then by Theorem 4.1.7

G = x(G, F) . On the other hand, by using the construction of

Theorem 4.2.5 with the assignment V satisfying [Yt,y.,v-] = t i j

(y+>Y-,»Y-)t i j t the associator in G , we obtain the loop (S^,* ) which is isomorphic to x(G, r) (and hence to G) since the function |x of Theorem 4.2.5 is the same as the function v of Theorem 4.1.7 *

We are now able to give a concise description of Gg^ •

(Recall that Gg^ was defined in §6 of Chapter I as the c-ty exp-3 loop corresponding to Hall's original HTS hg1 .)

COROLLARY 4.2.15: G8l =(S3,*v ) where (y^YgjY-,/-

[YgjYgjY^ “ h and A = ((hj) is a cyclic group of order 3 *

Proof: By Theorems 1.2.6, 1.5.16, 1.5.20, 2.2.4, 4.1.7, and 4.2.5 we know that °81 is the unique c-7% exp-3 loop of nilpotence class 2 and dimension 3 and furthermore that Gg^ 158

can be freely constructed as the set

e * e3 e2 ei ((h ,y2 ): 0 < e 'je1^e2»e3 £ 2i provided with the

e2 e2 q e2 2 e2 1 operation *v where (h ,Yg ' , Y2 * , ' ) *v

ei ei ^ ei 2 el 1 e2 S1 + ^ ^*2 ^*1 Ch S y ,1'3^ ' ^i'1) = (h ^ 1 Sjl ^o^Ya^Yi)

where (y3,Y£>Y-l) = [v3,Y2,Y1] = h = (y^ Y ^ Y - ^ • □

In fact, as an illustration of the remarks following the

proof of Theorem b.2.^ we see that Gg^ can be even more simply

constructed as GF(3)^ with the operation ^e2je2J3>e2,2,e2,l^ * (ei*el,3,el,2'el,l) =

(e2 + ei +^,i(3#2>l)je2^3 + ei,3'e2,2 + el,2'e2,l +el,l) * Wote how well suited this rule, and more generally any product rule as defined in Definition ^.2.3, is for a computer program. I have written a program which prints out the 1080 blocks of the historic design •

We can now give a direct verification that Wg^ fails to satisfy the ’’three dimensional" axioms AS-31 and AS-3 for an affine space given in §3 Chapter I. Recall that the triples of Mg^ are given in terms of the operation * of Gg.^ by x * y i= x-^ * y~^ . We temporarily adopt the following notation:

— — — 2 let e = e , p = q = y2> r = Yg* and x = x *x for all 2 elements x of Gg^ . Consider the lines 1^ = [q,q*p,q*p J , 2 2 - te,p,p } , and =■ (r,r *p,r *p ] , is "parallel" to

since <1^ U I^> = {e,p,p2,q,q *p,q *p2,q2,q2 *p,q2 *p2) is 159

a plane and L, is "parallel" to since (l*> U L^) = 2 '2 2 2 2 2 te*PjP >r,r*p,r*p ,r ,r *p,r *p ) is also a plane. However, we shall see that and are not coplanar and hence are not "paraHel.,, The nine points of the plane (L^ U {rj ) are r,q,q *P,q *p2,r « q = r2 *q2,r o (q *p) *= r2 * (q2 *p2),r « (q *p2 ) = 2 2 r * (q *p) , and the two additional points q°[r2 *(q2 *p2)] = q2 *[r*(q*p)] = h^0- 1 ^ 2 *1 - 0 '1)*(r*p) = h * (r *p) and r»[h*(r*p)] = r2 * [h2 * (r2 *-p~ )] *= h2 * (r *p2 ) •

If I>^ and were coplanar then certainly = U trj) but this is not the case since r *p ^ U [r] ) . Also note that the planes ^1^ U ) and U (rj) intersect only in the point r .

§3* The Construction of Loops with Specified Properties

We begin by constructing a c-7% exp-3 loop of nilpotence class 2 for each possible cardinality of the associator subloop,

THEOREM U.3.1: If G is a cexp-3 loop of nilpotence 1 class 2 and dimension d > 3 then 3 < jG'f < 3 . j Conversely, for all d > 3 and for all a with 1 < a < (^) there exists a c-7% exp-3 loop, G , of nilpotence class 2 such that d(G) = d and |G'| = 3& •

Proof: First assume that G is a c-ty exp-3 loop of nilpotence class 2 and dimension d > 3 . Let f « (Yd*Yd l6o

be an ordered basis of G . By Corollary 4.1.8, G 1 is generated

by the set of (^) basic associators

t(Yt,V±,Yj): l

one of these basic associators is not equal to the identity „ (?) element of G and hence |G’ | > 3 i and certainly |G' f < 3

since G r is an elementary abelian 3-group. Next assume that

d > 3 and a with 1 < a < (d) are given. We use the

construction of Theorem 4.2.^ by letting A be the elementary

abelian 3-group of dimension a with basis (hphg, .. .,haJ

and then defining the mapping V as follows. First we

lexicographically order the set of ordered triples

((t,i,j): l

iff t < t' or t = t r and i < i* or t » t', i = i*, and

j < j ’ ; second we consider the unique order preserving map

{(t,i,j): l a . By Theorem 4.2.5, (S^, *y) is a c-71\ exp-3 loop of nilpotence class 2 and dimension d . Furthermore, the associator subloop A is generated by the set of a independent basic associators [ [ Y^j Yj ]: 1 < (t,i,j)^ < a) . Hence,

K Sd^V>’l = 3 a . □

Notice that when a = (d) in Theorem 4.3*1 we have constructed the free c-77? exp-3 loop of nilpotence class 2 and dimension d . 161

DEFINITION 4.3*2: A c-JT; exp-3 loop is said to "be

irreducible iff it is not isomorphic to the direct product of

■two non-trivial c-771 exp-3 loops. We shall denote the free c-#J

exp-3 loop of nilpotence class 2 and dimension d by Fg ^ .

COROLLARY 4.3*3: For all d > 3 , |Fg d | = 3 d+(3} and

F~ , is irreducible. 2,d

Proof: It is immediate from the construction of Theorem

d+d) 4.3.1 that |Fg d | « 3 * Suppose that Fg d = ^ ® Hg

where dim(H^) = d^ with 1 < d^ < d . By Proposition 2 .3.17

dim(Hg) = d - d^ ; also nil(H^) , nil(Hg) < 2, so by Theorem dn d-d. ( ) C ) 4.3.1, |H^| < 3 ^ and lH£ | < 3 ^ * Again by Proposition d7 d-d. (,)+( o ) 2 .3.17, 1(1^®!^)'!= iHi i |h^ I < 3 J J and we would have d d - d d d(Fg 3 ^*^(3 ^ (since there are 3 element subsets of a d element set which do not lie entirely within a proper d^ element subset or its complement) which is a contradiction.

Hence, Fn , is irreducible, □ ' 2, d

The irreducible loops are the basic building blocks for constructing new loops using the operation of taking the direct product. The irreducible loops of nilpotence class at most 2 which we have seen so far are C^, Gg.^ = Fg ^,Fg ^,Fg .

After noting some properties of irreducible loops we shall construct 162

another Infinite family of them. For the next two propositions

G is assumed to be a cexp-3 loop of nilpotence class 2 and

dimension d > t . (We have full knowledge of Gg^ which is

the only such loop with dimension d = 3 *)

PROPOSITI OH ^.3.^: If v € Z(G)\G* then for some H,

G = <(y)) ® H ; that is, the cyclic group of order 3 generated

by y is a direct factor of G .

Proof: By Proposition 2.3*2, y is n°f a non-generator of

G, so there exists a basis r = tY*Yd i'****Yi^ of G . We

consider the ordered basis r = (y,Yd by Theorem

1+.1.7, G = x(G,f) . Let H - ( (g, Y^d_^1,..., Y^1): g € G' and

0 < e^ < 2 for all i f 1^ provided with the operation of Theorem ^.2.5 where [y+>Y->Y-] = (y+,Y-,Y-) » "the associator “ i J t l j in G , for all values of t, i, and j with l

(Since d - 1 > 3 , H is a c-^j exp-3 loop of nilpotence class 2.)

We shall show that the bisection ijr: x(G, r) ■* {(y)) ® H

f 6(1-1 6l\ / ( 6(1-1 & 1 \ \ (B,Y ,Yd _! i •••jY-l ) (Y »(8>Yd _1

d ^ d i i is an isomorphism. Given X£ = (g^Y * , Yd *± ",***, Y1 ' ) and

/ Gl,d 8l,d-l ei,lx . i _ = (g^jY ' JYd-l )r by Theorem ^.1.7 *x^^ «

fd fd-l fl (g2g-L^(E2,E1),Y /Yd _^ ,...,Y1 ) * But since y € Z(G), vfEg^) =

% t(t,i,j; TT (Y 1

(Y

1 < j < j . u . — j .

e2 ,a -i d -1

COROLLARY 4.3.5: If G is irreducible then Z(G) = G' .

Proof: Since nil(G) = 2, G ’ e Z(G) . And by Proposition

4.3.4, Z(G) s G' ; hence G' = Z{G) . □

Note that by considering the loop Gg^ & Gg^ we see that the converse of Corollary 4.3*5 is false. The following result

is an immediate consequence of Corollaries 4.3*3 and 4.3.5 .

COROLLARY 4.3.6: Z(F2 d ) «= d .

COROLLARY 4.3*7: If G is irreducible then [G: Z(G)] = 3d

Proof: By Theorem 2 .3.6 and Corollary 4.3*5, [G: Z(G)] =

|G|/|Z(G)| = lG(/|G’| = 3d □

Again, the loop Gg^ <55 Gg^ shows that the converse of

Corollary 4 .3.7 is false. 104-

FROPOSITIQN k.3.Q: If|z(G) | = 3 then G is irreducible.

Proof: We prove the contrapositive of the statement. If

G = Hx K Hg then |z(G) | = Izfj^) | |z(l^) | > 32 by the upper

central nilpotence of and . □

So in light of Proposition ^.3*8, our goal will be to

construct c-ty exp-3 loops of nilpotence class 2 which have

center of cardinality 3* These willenable us to complete the

table given in §3 of Chapter II. We shall be concerned more

generally with the construction of c-ty exp-3 loops, G , of

nilpotence class 2 satisfying G ’ = Z(G) . In using the

construction of Theorem ^f.2.^, necessarily A = ( 1 c ZffS^, *y))

and we must choose the assignment mapping V so that the

elements of A are the only elements of Z(( ^ )) . The

following lemma provides a complicated but useful arithmetic

criterion for the numbers q . .(£) (defined below) to t} 3 satisfy in order that A = Z((Sd,*y)) *

LEMMA ^4-.3 .9: If A is the elementary abelian 3-group

® ® ... & with basis

( ( (® j _ i* ® j * * * # ® ) t ' *m > ® f •** j ® t )) j that is

A = ((h^^h^*^1^ . . . ^ 1)): 0 < q(£) < 2 for all I 6 IR) ;

if the assignment mapping is

V: B -*■ A and if

(yt,Yi,Yj) ** [ Yt/ Y±/ ,..^1^ ' ’3 ) ; 165

(S^,*y) is the loop of Theorem 4,2.5 (recall that by definition

Vv * A = (B ) c A , so we can assume that [h^,!^, ...,h^} has been

chosen so that for all £ € I there exists at least one set of

values (tr,il,t}'} such that q , ,, .,(£) = 1 and t; , 1 , j ± . j.(je') - o for v t I) ; then A = Z((Sd,*y )) iff

the following situation holds: if (e^e^ i#* " ,ei^ ^ GF(3)^

satisfies the property that for all r and s with 1 < s < r < d

and for all £ f I K. , t £ o in. r toU)e s m + 2 £ m m + d>m>r ' ' r>m>s * 7

1 Q = 0 then e = 0 for all m £ I, . . r f s f m tn m q s >m> 1 7 7

Proof: To quote Marshall Hall, "The proof is not much longer than the statement," By Theorem 2.2.10 an element

ea ei x = ^&jYd >*'*'Yi ) of (Sd,JV satisfies x €Z((Sd,*v )) iff for all r and s with 1 < s < r < d, (x,y ,y ) = e * . _ _ — r s g 0 6 6 Since x = a *[ydd * * (-.-(y22 ))1 by Lemma 4.2.14, we know that x € Z((Sd, *y)) iff for all r and s with

1 < s < r < d, e* = " (YffiV r,Vs) = TT ( Ym, Yr, Yg ] m * m = l d >m > r

2e e TT [V »Y ,Yj m * TT [V ,Y jV ] m iff for all r and r>m>s. . r nr 's s>m>l^ r' s* 'm s with 1 < s < r < d, (h^,h^-:L,...,h°) =

TT (l^n'r'Sm > r

2q, (k)e 2q_^ (l)e TT “ ) * r >m > s 166 q w (k)e q (1)m > 1

with 1 < s < r < d and for all £ 6 1, , 0 = £ a (£)e k h 'j.m -> Tn,r,a' m

2 E Vm,s^)em+ E Vs.m^m* And to require that r >mm> s * * s>m>l * *

such an element x, contained in Z( (S^, * )) , be an element

of (S^,*v )' is precisely to require that em = 0 f°r all

m f Id , . □

THEOREM U.3.10: for all d > 3 with d / !* , there exists a c-2?; exp-3 loop of nilpotence class 2, R^ tsuch that d(Rd) = d, |Rd| - 3 d + 1 , and |Z(Hd)| = 3 .

Proof: We shall use the construction of Theorem k.Q.lj .

Necessarily, A must be the cyclic group of order 3 and we write A = {h^: 0 < q < 2} . If we can choose the assignment mapping V so that the arithmetic condition of Lemma U.S.9 is satisfied then Rd = (Sd, * ) will satisfy d(Rd ) = d,

|Rd| = 3d + ^, and |Z(Rd) | = 3 ■ In the notation of Lemma U*3*9 we have abbreviated h^ by h , and we shai 1 also abbreviate q. . j(l) by q . . The proof will be complete when we have established the lemma that follows. We merely set Yj = Qt i -J [Y^,Y^jYj ] = h ’ where the numbers q^ ^ j are the specially assigned numbers given in the lemma. □ 167

LEMMA ^.3*11: For all d > 3 with d / 4 , there exists an

assignment mapping Q : B -* GF(3) such that if for all r and s

with 1 < s < r < d, £ d > m > r

q e = 0 then e = 0 for all m 6 I r, s,m m m

Proof: We shall show that the conclusion is valid for d = 3 *1 >= 3, d - 3 *1 + 2 = and d = 3 .2 + 1 = 7 * we then show by induction on u that the conclusion is valid for d = 3 • u with u > 2j d = 3u + 2 with u > 2, and d = 3u + 1 with u > 3 •

The case d = 3 (note that this is the case of

equation

(3,2)

(3.1) = 0

(2.1 ) 0

Consider the assignment ^3 2 1 ” ^ : el = ^e2 = e3 = ^ that e = 0 for all m € . m 3 The case d = 5 : 168

i£jJ equation

(5,^ *b,1l,3e3 + S>,2e2 + S,^,l*l 0

(5,3 2q5,4,3e4 + ^,3,2^ + ^,3,1*1 = 0

(5,2 aq5 , ^ 2 % + 2q5,3,2e3 + = °

(5,1 2q5,4,le4 + 2q5,3, le3 + 2q5,2,le2 = 0

(M V * , 3 e5 + qS 3 , 2 e2 + % , 3,lel " ° (4,2 q5,^,2e5 + 2q^,3,2e3 + q4,2,lel = 0

(4,1 + 2q4,3,le3 + 2q4,2,le2 = 0

(3,2 S,3,2e5 + q4,3,2e4 + q3,2,lel = 0

(3,1 q5,3,le5 + q4,3,le4 + 2q3,2,le2 = 0

(2,1 S,2,le5 + q4,2,le4 + q3,2,l e3 = °

Assign ^ 3 = 1 and = *^,4 l = 0j equation (5,*0 forces = 0 . Assign ^ 3 2 = % 3 1 = equation (5,3) forces e^ = 0 . Assign 3 2 = ^ 3 : = 0 5 equation (4,3) forces e = 0 . Assign qq p . = 1 ; equation (3,1) forces e2 = O and equation (3,2) forces e = 0 . The numbers q p and q. may be assigned arbitrarily; to be specific 5,£=,.l a,c;,± let S ,2,l = q4 ,2,1 = ° *

The case d = 7 :

Of the (7>) = 2 1 equations we list only those that are needed to achieve our goal. 169

(r*s equation______<7,6 *7,6,5*5 + *?,6,i+eU + **7,6,3*$ + *7,6,2e2 + *7,6,lel = 0

(7,5 2*7,6,5*6 + * 7 , 5 , ^ + *7,5,3*3 + *7,5,2*2 + *7,5,1*1 = °

(6,5 q7,6,5e7 + *6,5,-^ + q6,5,3e3 + *6,5,2e2 + q6,5,lel = 0

(7,3 £q7,6,386 + 2q7,5,3e5 + ^ >^,f^ + *7,3,2e2 + *7,3,1°! = °

(3,2 q7,3,2e7 + q6,3,2e6 + ^,3,2% + %,3,2eh + q3,2,lel = 0

(3,1 **7,3,1*1 + q6,3,l86 + %,3,l*lj + ^,3,1^ + 2*3,2,le2 = 0

(2,1 *7,2,1*7 + *6,2,le6 + *5,2,le5 + qi;,2,leU + q3,2,le3 = °

Assign *7,6,5 = 1 and * 7 , M = *7,6,3 = *7,6,2 = *7,6,1 = °> equation (7,6) forces = 0 . Assign = q^^_2 =q^^^ = 0j equation (7,5) forces e^ = 0 . Assign

“ q6,5,3 - %-j,Z " q6 , ^ l " 0 5 e forces e^ = 0 . Assign q^, }+ ^ = 1 and ^ 2 = ^ 3 1 = Oj equation

(7,3) forces = 0 . Assign q^ g -^ = 1 ; we have = e^ = e^ = e^ = 0 so equations (3,2), (3,1), and (2,1 ) force e. = en = e_ = 0 . Hence e = 0 for all m € I_ , We have 1 2 3 in l made 17 of the (^) =35 assignments, and the remaining 18 may­ be made arbitrarily. To be specific we set them all equal to zero.

Now we proceed with the induction by assuming that any of the following holds: d = 3u with u > 2, d = 3u + 2 with u > 2, or d = 3u + l with u > 3 ; and that the conclusion of 170

the lemma is valid for d = 3(u - 1), d = 3(11 - 1) + 2, and

d=3(u-l) + l.

We begin by making the following assignments:

qd,d - l,d - 2 = 1

and

q^ d -1> J = qd,d -2, j = qd -l,d -2, j “ 0 for 1 - ^ - d - 3 *

Hence, in equation (d,d-l) we have

0 = r q e = e in equation (d,d-2) we have d -1 > m > 1 3 3

0 - 2(3 d,d-l,d-2j a 1 a neA d-1 i + d _3 ^>m>1 Md,d q, , -2,m „ e m = 2e, d - 1 implying that e^ ^ = 0, and in equation (d-l,d-2) we have

0 = Q d.da a -t l,da - 2 n ed A + 0E r 1 <1d-l,d-2,mma t a o e “ d ■ So far we 7 7 d-3>ni^>X 7 7 have forced e , = e, = e 0 = 0 . We now consider the subset u a - 1 a — c- of our set of equations having labels (r, s) with l

(r,s) Z q_ e + 2 £ q e + Z d e = 0. , ^ . Tn,r, s m ^ . rr.m, s m ^ , T.s.m m d>m>r 3 3 r>m>s 3 3 s>m>l 3 3

Since e^ =1 ~ ed 2 = ^ 3 ^ese equations simplify to

(r,s) Z Q e + 2 £ q e + d^ -3 >m___ > r„ Tn,r,s m r>m>s x*,m, s m 77

Y q e = 0 and by induction there is an assignment . , r.s.mm s>m>l ' '

{^. 1 < j < i < t < d - 3 } which forces ed_^=ed_1+ = ...= ep = e = 0 . We arbitrarily assign q = 0 for t € {d, d - 1, d - 2) and 1 < j < i < d ” 3 * and the proof is complete. □ 171

The next result is part of Theorem 2.3.20 which is due to

Beneteau. I believe that the proof given here yields more insight into the situation.

PROPOSITION 4.3*1^: There does not exist a c-^ exp-3 loop,

G, of nilpotence class 2 satisfying |g ( =3^ and |Z(G)| = 3 *

Proof: As mentioned in the proof of Corollary 4.2.1 5 , Gg^ is the unique c-7l\ exp-3 loop of nilpotence class 2 and dimension

3 . Hence, if such a loop G were to exist it would satisfy d(G) > 4 , and in fact d(G) = 4 by Proposition 2.3*12 . We luiow by Theorem 4.1.7 that any such loop could be constructed by the method of Theorem 4.2.9; hence by Lemma 4.3.9 (using the notation of Lemma 4-3*11) such a G exists iff there is an assignment Q : B ■> GF(3) such that the (^) = 6

- qt,i,j equations below force e^ = e^ = = e^ = 0 ,

(rjs) equation

(J+,3) = 0 q4,3,2e2 + q^3,lel = 0 (M) 2 q^,3,2e3 + q4,2,lel = 0 (M) 2q4,3,le3 + 2q4,2,le2 (3,2) = 0 q4,3,2e4 + q3,2,lel (3,1) = 0 q^,3,le^ + 2q3,2,le2 (2,1) = 0 q4,2,le4 + q3,2,le3 172

We temporarily adopt the notation g " a» Q ■4,3,1 - To show that no such G exists, *4,2,1 = Y' ^ q3,2,l ■ S • it suffices to show that the matrix

0 0 GL P 0 2a 0 Y 0 2p 0 M *= 2y 0! 0 0 5 p 0 2b 0 Y 6 0 0

has rank less than U . (Of course the entries of M are elements

of GF(3) ■) We shall indicate the row operation of replacing row

i by c • (row j) + (row i) by writing E. R, + cR . , Performing i J such row operations on a matrix does not change the rank of the

matrix.

Case 1; a = 0 Here

0 0 0 P 0 0 0 Y 0 2P 2 Y 0 M = 0 0 0 5 p 0 2b 0 Y b 0 0

If p = 0 then

0 0 0 0 0 0 0 Y 0 0 0 M = 2y 0 0 0 5 0 0 25 0 Y 5 0 0 which has rank less than ^ . If p ^ 0 then 173

0 0 0 ------► 0 0 0 M -1 P -1 P 1*2 h- 1*2 - p VR-l 0 0 0 0 0 0 0 0 RA00 *"* + P &Ro j 0 2 p 2y 0 0 2p 2V 0 Hi. Ri, - P_1&R 0 0 0 0 0 0 0 0 p 0 26 0 p 0 26 0 -1 „ 0 0 0 0 e6 ”• R6 ' b"1yE5 ° 6 P Y6 0

which has rank less than 4 .

Case 2: a 0

M 0 0 a P 0 0 a p 0 2a 0 R3 + a'1YR1 0 2a 0 R3 ~ R3 _a~1PR2 Y Y R3 0 0 0 0 0 0 2y 2a_1py R*. + a"16R1 a 0 5 R^ ~ R^ - a-1pR4 a 0 0 6 0 0 0 0 0 0 26 R6 ^ R6 - a_1YR4 0 -a_1p6 Eg + a-^Rg 0 0 0 0 0 6 0 -a_1y6

which has rank less than 4 . □

Given the negative result of Proposition 4.3.12, we do the next best thing.

THEOREM 4.3.13: There exists an irreducible c-ty exp-3 loop,

K, of nilpotence class 2 satisfying d(K) = 4, |k | = 3^, and

|Z(K)| = 32 .

Proof: We again use the construction of Theorem 4.2.9 . 2 necessarily, A is the elementary abelian 3-group of order 3 ;

A = { ( h ^ ^ ^ h ^ 1^): 0 < q(2),q(l) < 2) . By Lemma 4 .3.9 if

V : B ■* A can be defined so that the % i % 1 (W V ~ (V 'V } 17 U

arithmetic requirement of Lemma ^.3-9 is satisfied then

K = will have the stated properties. Consider the

following equations for I £ {1,2) :

= 0 q4,3,2^e2 + qM * i ^ ei

£q^,3,2^^e3 + ql^,2,l^^ei = 0

= 0 2%,3,l*A)e3 + 2qtf,2,l^^e2 (3,2)^: qU,3,2fi)e4 + q3 , 2 , i ^ ei = 0

= 0 (3,l)x: qi+,3,l^ej+ + 2q3,2,l(^ e2

= 0 (2,l)x= qU,2,l^^el+ + q3 , 2 , l ^ e3

Assign q. , p(l) = q. _(2) = 1 and assign all the other numbers q, . .(jE) to be zero. Equation (^,3)o becomes e = 0 ; equation t,l,J d. ± (J+,3)^ becomes e£ = 0 ; equation (k/2)^ becomes 2e3 ~ o implying = 0 ; and equation (3,2)^ becomes e^ = 0 , Hence, this assignment forces = 0 for all m € and K = (sij.j*y) has the desired properties. □

We remark that using this technique many other irreducible loops with centers of cardinality larger than 3 can be constructed.

I believe that in the future this will be an important part of extending the catalogue of "small" HTS's beyond what is done in

Chapter V of this treatise.

We can now make good our promise of completing the table of

§3 of Chapter II. 175

THEOREM 4.3 * 14: Given any ordered pair (3Z,3m ) / (3X,35 )

with m > 1+ and 1 < z < m - 3 } there exists a c-ty exp-3 loop,

Gz m * °** nilP°‘tence class 2 satisfying |Gz | = 3™ and

\Z<-az,m> I ' ^ ‘

z m Proof: Given such an ordered pair (3 ,3 ) t there are two / 7 — 1 cases to consider- If z * m - 4 then let G = R ® C„ ' z,m m - z 3 where R is one of the loops constructed in Theorem 4-3*10 , m - z Certainly nil(GzJ = - 3»-,+ 1 .3,-1 _ 3» and |Z(Gz

3 • 3Z ^ = 3Z * We point out that in this case d(Gz m ) =

(m-z)+ (z-l) =m-l and jG' | = 3 * If z = m - 4 let 2 — O G = K ® ^ where K is the loop constructed in Theorem z,m 3 4*3*13. We have nil(G^m ) = 2 , Jg^ J = |k ||c“ " 2 | =

36 *3Z “2 = 3Z + ^ = 3®, and |Z (G ^ ) 1 = jz(K) | ]z(CZ " 2)) =

3^*3Z”2 =3Z* In this case d (Gz ^ ) = 4 + (z-2) = z + 2 = m - 2

“ d K, ml=3a . □

We remark that other infinite families of loops can be constructed as well: for example, (K ® R^ ® m € N) ,

[K « R5 ® R6 x C^: m f Nj, etc.

Because of the relation of c-ty exp-3 loops to their corresponding rank 4 PMD's, we deal almost exclusively with finite loops. However, we can certainly construct many infinite c-ty exp-3 loops in addition to the Bruck loop R and the Malbos loop M . (Bruck proved in [6] that ]z(B)| = 1 and it is also 176

true that |Z(M)[ = 1 .) For example, we have

® C3 ® C3 & * • *, ® G8i ^ G8l ^ * * * * ®3 ® ®i|. ® ® * * *) etc.; each of these loops has a non-trivial, in fact infinite,

center. We close this chapter with the construction of an

infinite c-ty exp-3 loop of nilpotence class 2 with center of

cardinality 3 (which therefore can not be a direct product of

finite loops). The idea that a "locally finite" structure could

be used is due to D. K. Ray-Chaudhuri.

DEFINITION 4.3-15: Let A = (hq: 0 < q < 2J be the cyclic 1 * o group of order 3 generated by h = h and write e for h ;

for all i £ N let = (y°, Y^jY^J be the cyclic group of order 3 generated by (again we write y^ for y^ and e for y.) ; let T = A x X G. ; let S = [(a,y ,y0 ,... ) £ T : X OO i 1 00 J- £, CD

(i: ei / 0) is a finite set); for a £ A, let a = (a,e,e, ...) and

A = (a: a £ A) j and for k f (0,1,2) let y!^ = (e*,yk,e,e, ... ),

k k V2 *= (e y2,e,...), ... . For all u £ N let

^ y3u' Y3u - V y3u - 2} = ^ an(i ^or a"^ °^her values of t, i, and j 1 * with t > i > j let Iy+.jY-jY.J = e and we define a product, 1 J * . on S as follows: given elements x~.x. £ S with o o 1 CO 6 6 0 6 Xg »= (®2 J f Yg f • • • ) and x^ = (a^, y^ f f y^ f let f f 1 2 x2 * xl c ta2ai^fE2jEl^Yl * Y2 >’" } Where M'C^2,El) " % TT [Yj-^Y-#Yj3 * * Note that by the local finiteness 1 < j < i < t 1 3 177

of S , p,(E0,E, ) is a finite product. For u € N we define oo c X 6 6 Ju *= ((a, Y-j1, Y22, * • *) e S^: ed = 0 for all d with d > 3uJ .

PROPOSITION U.3.I6 : (S^*) is an infinite c-Vt exp-3

loop of nilpotence class 2 satsifying Z((S ,*)) = A j that is, 00

lz((s«.*>>l - 3 .

Proof: We refer to the proof of Lemma U.3*ll (in particular,

the case d = 3a) and note the key fact that the associator

symbols of (S^ *) have been defined so that for

all u £ N, (Jyj*) is isomorphic to the loop R^u . An

isomorphism is t|r: (Ju#*) -*■

/I 3u \ /„ 3u lv (a7 Vj j • * *} e> • * • ) (a> V^u > • * * j )

Since every finite set of elements of (S^ *) is contained in

some J , it is immediate that (S , *) is a c-?/J exp-3 loop XI tjer with (S^*)’ = A which satisfies the simplified expansion law.

For example, to show that (S 00,*) satisfies the ^identity, given elements x ^ x ^ x ^ € S we consider the smallest value of u such that x^,Xg,x1 f J^: since = R , x^ *[*3 * (x^ *xd )] =

(x^ *Xg) * (x^ *x^) . Since for all u £ N, Z((Ju,*)) = A , we have Z((S , *)) c A ; and by definition of the operation *, 00 A s Z((S^*)) . Hence, (S^ *) has nilpotence class 2 and

|Z((S co,*))| = 3 . C CHAPTER V

THE CATALOGUE OF HALL TRIPLE SYSTEMS OF

CARDINALITY AT MOST 3^

4 ^ § 1. General Remarks and the Cases |G | - 3 and fG | = 3

We shall analyze and catalogue the "small" HTS's totally

in terms of their coordinatizing loops which are given in the

correspondence of Theorem 1.5*16 .

As we have seen in §3 of Chapter IV the freedom we have

in the construction of Theorem 4.2.5 is extremely useful in

constructing many c-7ft exp-3 loops of nilpotence class 2 .

However we must pay a price for that freedom: although a

given assignment mapping V certainly determines the loop

(S^,*y) constructed in Theorem 4.2.5, it is possible that a

given loop can arise from different assignments. We shall give

an example of this shortly. Furthermore it can be difficult

to determine whether a given loop is irreducible; for example,

I do not Know yet whether the loop , constructed in §1 of

Chapter III and used in the construction of the loops

Rfm^jiiig,!!^) , is irreducible. An even more difficult question

is whether there is a unique factorization of every c-7^ exp-3 loop of nilpotence class 2 into irreducible loops. I believe

178 179

that this problem deserves attention and should be studied in

the near future.

If we were allowed to assign all the basic associator -Jf. symbols to be e , that is [ y^t y^ ] = e for all t, i, and j with l

Theorem 4.2,5 would just yield the elementary abelian 3-group d * C3 since we would have A = (e ) . In view of this one might be tempted to think that the greater the number of basic T * associator symbols LY.i-,Y*,Y-J which are assigned to be e , t 1 j the closer the loop (S^, *y) will be to being a group. The following example shows that this is by no means the case.

Let d s* If and consider the assignment V given by

[Y3^Y2,Y1] = h and [Y^YyYg] = tY^,Y3,Y-L] = [ Y ^ Y g ^ ] = e*: here A = (fh]) . In the loop , <{y^,7g, y1) > = e e2 {(h *e,Y^ ,y2 ,y^ ): 0 < q,eye^te^ < 2 J and therefore

(p^Pg,^,^) is a basis of (S^,*v ) where p± = y± for i € I3 and = Y^ * (Y3 * (Yg * Yj_)) • By considering the ordered basis f* = (Pjf^P3jPgJP1 ) , we see by Theorems 4.1.7 and 4.2.5 that (S^,*v ) = x((Slt,*v ),Tr) = (S^,*v ,) where the assignment

V' is given by

[Y^jYgjY-^] = (P^jPg,Pj) “ tY3,Y2,Y1) = h

[ W Yl] * = (VYg^J^Yg^) = e * • h = h

C VY 3,Yx] = (P^Pg,^) *= (^,73, ^ X 72,73,^) = e* .h"1 * h2 and 180

f Y^,Y3,Y2] “ (3jj.#03>P2) = ^2 ^^1* ^3* ^2 ^ = ® »h = h .

Given a c-7% exp-3 loop, G , of nilpotence class 2 with

ordered basis f = (vd,Yd _ ±,. . Yx) J since G = x(G, r) , G

is completely determined by the ordered (^)-tuple of basic associators ((y^Y^Y^ )) where 1 < j < i < t < d . The listing of the elements of the ordered (^)-tuple is done according to the decreasing lexicographic order on the ordered triples (t,i,j) of subscripts; that is (t, i,j) s> (t',i',j') iff t > t* or t = t* and i > i* or t = t ’, i=i', and j > j ' • For example, when d = b we have the ordered -tuple

(C Y^j Y3, Y2), (Y^ y3, y^, (Y[|> Y2, yx), (Yy Y2, Y^ ) • However, we have just seen an example in which two different ordered bases of the same loop G yield distinct ordered (^)-tuples of basic associators.

DEFINITION 5-1.1: If G is a c-^ exp-3 loop of nilpotence class 2 with ordered basis r = (y^j * • * * Y-j_) (necessarily d > 3) then the a-vector of r is denoted by q(r) and is defined to be the ordered (^J-tuple of basic associators ordered as described in the preceding paragraph; that is, a(r) = ((Y^*Y^>Y^)) *

If A is an elementary abelian 3-group with identity element e then two ordered (^)-tuples, a = (a ,a ,...,a . ) and 3 1 2 (d} a' = (a£,a^,., .,a'd ), of elements of A are said to be equivalent V iff there exists a c-J7? exp-3 loop, G , of nilpotence class 2 with l8l

ordered bases r and f7 such that a(f) <= Qt and a d 77) = a' .

When a and a' are equivalent we shall write a ~ a* . The

ordered (^)-tuple (e,e,...,e) is said to be trivial.

j If l

ordered (g)-tuple, a, satisfies a =a(r) for an appropriate

choice of G and r . Hence, a loop G corresponds precisely to the equivalence class of the a-vectors of its ordered bases.

Our method for obtaining the exact list of non-isomorphic c-77\ exp-3 loops of nilpotence class 2 and a given "small" cardinality is to determine, for the relevant group A = G f , the set of equivalence classes of non-trivial ordered (g)-tuples. We begin with what we can now view as the easy case of ■ |G| i = 3 k •

PROPOSITION 5.1.2 (Hall); Gg^ is the unique non-associative Ij. c.-*n\ exp-3 loop having cardinality 3 .

Proof: Let G be a non-associative c-^ exp-3 loop of if / X cardinality 3 • > 3 since otherwise G would be a group.

By Proposition 2.3*12,we must have d(G) = 3 and by Theorem

2.3.8, d(G/G») *= 3, [g/g1 | = 33, and |G*| - 3 - (Note that

|G'| = 3 implies that nil(G) = 2 .) Hence G' = (h^: 0 < q < 2) and with r = (Yg^Y2jV-^) aa ordered basis of G we may assume w.l.o.g. that a(T) = (h) . The only other non-trivial 1-tuple 2 2 is (h ) ; and we see that P = (YgjY^Y^) is also an ordered 2 2 basis of G and since the associator (YgjYg^Y-^) = (Yg/YgjY^) 182

— 2 2 we have a ( r ' ) = (h ) . Hence, (h) ~ (h ) and there is at

most one non-associative c-V{ exp-3 loop of cardinality 3^ . And

in fact, in Corollary 4.2.15 we have constructed precisely

as the loop with ordered "basis ( , Vi) in which the

associator (YgjYpjY^) ** h where <(h}> » . □

In the next three propositions we shali give a much different

proof of Theorem 2.3.20 than Beneteau's original one. Note that

the part dealing with the case of nilpotence class 2 gives another

explanation, in addition to the proof of Proposition 4.3*12, of

the exceptional, case of the non-existence of a c-l7{ exp-3 loop,

G, of nilpotence class 2 with |g | =3^ and |Z(G) f = 3 .

PROPOSITION 5 .1.31 If G is a non-associative c-J7? exp-3 loop and |g[ =3^ then nil(G) = 2 .

Proof: If we had nil(G) > 3 then d(G) = d(G/G') > 4 implying that |G/G' | > 3^ * Hence we would have |G* j < 3 forcing |G2| = 1 which is impossible. Hence nil(G) < 2 and since G is non-associative, nil(G) = 2 . □

We remark here that Proposition 5.1.3, Proposition 4.3*12, and

Theorem 2.3.19 can be used together to immediately derive

Theorem 2.3.20 .

PROPOSITION ^.1.4; If G is a c-ty exp-3 loop of nilpotence class 2 with |g| = 3^ then d(G) = 4 . 183

Proof: If we had d(G) = 3 then by Theorem U.3.1 we 1. would have )G1 | = 3 and |G| = 3 which is not the case.

By Proposition 2.3.12, d(G) < 5 and if we had d{G) = 5 then

we would have nil(G) =■ 1 which is also not the case. Hence,

'a(G) = i* . □

PROPOSITION 3-1.5: = G8l « C3 is the unique c-ty r exp-3 loop of nilpotence class 2 and cardinality 3 •

Proof: By Proposition 5 .I.U a c-ty exp-3 loop G of R nilpotence class 2 and cardinality 3 necessarily satisfies d(G) = 4 and |G' | = 3 * So in terms of the discussion following Definition 5.1.1, we must determine all the equivalence classes of non-trivial ordered lf-tuples of elements of the group G 1 = {hq : 0 < q < 2} , Furthermore, we may assume w.l.o.g. that G has an ordered basis r = Y2> Y-j_) satisfying

(Yj^YyYg) = h . In particular let V = (y^, y£, YjJ be an ordered basis of G8l ® where (Cy^>Y3#Y2}) - Gqj ® te] and

<(y^)) = (e) ® • We have a (v) -

((Y4^Y3jY2 )j (Yif,Y3>Y1),(Yi|>Y2,Y1 ), (Y3,Y2,Y1 )) = (h,e,e,e). We shall show that every non-trivial ordered k -tuple with first coordinate h (there are 3 = 27 of them) is equivalent to a(V) . Given e. ,e ,e£ € (0,1,2) consider the following ordered G G 6 basis of G8l ® C3 : v(e^,e3,e2 ) = CY1+,Y3,Y2,Y^(Y33(Y22Y1))) •

(Note that v(0,0,0) - v ,) By the simplified expansion law

, Go (SE), a(V(e^,e 3 ,e2 )) B ( C Y^j Y3 * Y2 )> (Y ^ Y y Y2 ) (y^Y^Y-^, IS* e3 ek ep 2e- e, (YV v2,Y3 ) ( v y 2.y1), Cy3,Y£,Y^) (YgjYgjYj^)) = (h,h ,h 3,h ) . e2 % Hence, (h,e,e,e) - (h,h ,h ,h ) for all e^,e3,e2 € [0,1,2]

and the proof is complete* □

For the sake of completeness of this chapter we restate

Theorem 2.3.20 here.

THEOREM 5.1.6 (Beneteau): = G ^ (5? is the unique

non-associative c-ft; exp-3 loop of cardinality 3 *

§2. The Case |G I = 3 6

We first show that there is no c-/7f exp-3 loop, G ,

satisfying [G| = 3^ and nil(G) > 3 ; then we use the method introduced in §1 of this chapter to treat the case of nil(G) = 2 ,

That no such G with nil(G) = 3 can exist is by no means obvious, and in fact depends on the careful structural analysis of Theorem 5.2.5 . We begin by recording the following computational aid.

PROPOSITION 5*2.1: If G is a c-J5f loop and 2 € Z(G) then for all v,x,y € G, {wz,x,y} = (w,x,y); more generally

Z(G) = D(G) .

Proof: We see that (wx,y,z) = e * e *e = (w,y,z)(x,y,z) , so Z(G) = D(G) . (Recall Definition 2 .2.15.) Hence, 185

by Proposition 2.2.14, (vz,x,y) = (w,x,y) (z,x,y) = (w,x,y) •

(This can also be proved in a very straightforward manner by just

using the expansion law since (z,x,y) - ( (w,x,y), w, z ) =

((*,x,y),z,v) = e .) □

FROPOSITION 5.2.2: If G is a c-J7? exp-3 loop satisfying

[G | = 3 ^ and nil(G) > 3 then nil(G) = 3 and d(G) = 4 .

Proof: If d(G) > 5 then d(G/Gf) > 5 which would imply that 1G’j < 3 and that |G2 | = 1 which is not the case. Hence, d(G) < 4 and by Theorem 2.2.4, nil(G) *= 3 and d(G) = 4 . □

So, we shall now consider a c-ty exp-3 loop H of arbitrary cardinality satisfying nil(H) = 3 ■ By part (i) of Theorem 2.2.13

(H*, H’,H) £ = {ej , and in particular (H'jH'jH') * te] meaning that H* is an elementary abelian 3“group. We have

H r « with 5(H) > 2 and moreover Hg “= ,

H f = « C3 ^H^ = « C3 ^H^ where \(H) + e(H) = 5(H) and

\(H),e(H) > 1 .

DEFINITION 5 *^.3 : For a c-ty exp-3 loop H of nilpotence class 3 and dimension d with basis ^Y^Y^ i^***'Y1) the basic 1-associator (Y^>Y^>Yj) will be denoted by A(t,i,j) and the basic 2-associator ((7,,\ ,Y^)>Y.) will be denoted "C 1 J K A by B(t,i,j;k,I) . 186

Note that if k = l or if {k,X) ^ (t,i,j} then

B(t,i, j ;k, X) = e the latter following from part (v) of

Theorem 2.2.1 .

Next we consider the situation when d(H) « 4 . Let

f »V3^Y2j Yj_) be a basis of H . 1^ is generated "by the

following set of twelve basic 2-associators:

(B(U,3,2;4,1), B(4,3,l;4,2), B(4,2,i ;4,3), B(3,2,1;U,3),

B(4,3,2;3,l), B(4,3,l;3,2), B(1*,2,i ;3,2), B(3,2,l;l*,2),

8( ^ 3,252,1), b(J*,3,1;2,i), B(4,2,1;3,1), b(3,2,1jM)) .

Of course these associators together with the set of basic

1-associators (Aft,i,J): 1 < j < i < t < generate all of

H' * However, as we see from part (ii) of Proposition 2,2.11

there are dependencies among the basic 2-associators. Because

of its importance we list this result again here and include

an original proof which is simpler than Bruck's proof.

PROPOSITION 5.2.U (Bruck): If w, x, y, and z are elements of the c-77? loop G then ( (x,y,w),w,z) = ((y,z,w),w,x ) .

Proof: All elements of G which we shall consider will be elements of the subloop > ** ((w,x,y,z}) which satisfies nil(y) < 3 by Theorem 2,2,4 . Let Y = (x,w,yz) , By (E)

Y « [ (x,w,y)(x,w,z)] .t where t = (y,z, (x,w,y )}(z,y, (x,w,z)) £ i & r

>2 c Z(») . Now e = (Y,w,yz) = (Y,w,y)(Y,w,z) , the first

equality following from the fact that Y 6Z(((x,w,yz}>) and

the second equality following from Y £ y* « = y nil(y) - 2 s if nil(y) ^ 3 (see the paragraph following Proposition 2.2,16)

and Y € D(» if nil(%) < 2 * Hence, (Y,z,w) = (Y,w,y). Next

we see that (Y,z,w) «. ([ (x,w,y)(x,w,z)]t,z,w) * ((x,w,y)(x,w,z),z,w) =

((xJw,y),z,w)((x,w,z)Jz,v) = ((x,w,y),z,w) ; the second equality

holds by (E) and because t 6 Z(>) , the third equality holds

by Proposition 2.2.14 and because (x,w,z) £ >' c D(y) , and

the fourth equality holds because (x,w,z) £ Z(([x,w,z}>) ,

Similarly (Y,w,y) = ((x,w,y),w,y)((x,w,z),w,y) = ((x,w,z),w,y) .

Hence, ((x,y,w),w,z) = ((x,w,y),z,w) = (Y,z,w) = (Y,w,y) =

((x,w,z),w,y) *= ((x,z,w),w,y)_1 . So, ((x,y,w),w,z) =

((x,z,w),Wjy)-"*" and since this is a valid identity for any

elements x, y, w, and z we also have ((z,x,w),w,y) =

((z,y,w),w,x)-^ , Therefore we obtain ((x,y,w) ,w,z) =

((x,z,w),w,y)-:L = ((z,x,w),w,y) = ( (z,y,w),w, x)-1 -

i(y,z,v ),v,x) • □

Returning to our loop H satisfying nil(H) «= 3 and

d(H) = 4 , we can now summarize the dependencies among its basic

2-associators in Table 2 below. For basic 2-associators of H the equality of Proposition 5.2.4 can be written

(1) 188

and by switching the subscripts j and k we obtain the

identity

(2) ((WY^'VV c K v v V ' V V '

the left hand side of which involves a different one of the

generating set of twelve basic 2-associators than either of those which appear in equation (l). Note however that the identity obtained from (1) by switching the subscripts i and

v V ' ylelds no new information since it involves no new elements from the twelve element generating set and is equivalent to J ft SL -v 1 ((y.>Yj^Y.),Y,,*Yv,)-1 which is the same thing as (l) . Also 1 J JC Xr ft the identity obtained from (l) by switching the subscripts i and $, {{yy Yi,Yj(,),Yje,Yk) = ( (Yi# Yk, Y^), Y^, Y^ ) , involves no new generating basic 2-associators and can be derived from

(1) and (2) as follows:

<(v,j

((Yi.Y^vp.Y^Yj) •

So, given an ordered if-tuple (i,,J,k, &) of distinct subscripts, we need not consider the l+-tuples (k,j,i,je) and (j,i,k, i) » 189

Table 2: Dependencies among the Generating Basic 2-Associators in a c-??f exp-3 Loop of Dimension 4 and Nilpotence Class 3

Dependence in the i .1 k I B(i,j,X;4,k = B(j,k,£;I»i) Generating Set

4 3 2 1 b (i+,3,i ;i >2 = B(3,2,i;i,4) B(l+,3,1;2,1) -B(3,2,l;4,l)

4 2 3 1 B(l+,2,1;1,3 = B(2,3,1;1,1+) B(1+j2,1;3,1)_1=B(3,2,1;4,1)

1+312 B(4,3,2;2,l = B(3,1,2;2,1+) B(l+,3,2;2,1) c B(3,2,l;l+,2)

1+13 2 B(4,l,2;2,3 - b (1,3,2;2,4) B(l+,2,1;3,2) =B(3,2,l;l+,2)

1+ 2 1 3 B(4,2,3i3,l = B(2,1,3;3,1+) B(l+J3,2;3,l)-1=B(3,2,l;lf,3)

1+12 3 b (i+,1,3;3,2 = B(1,2,3;3,1+) B(i+,3,li3,2)_1 = B(3,2,l;i+,3)

3 2 1 4 B(3, 2,4;4, 1 = B(2,1,!+;!+, 3) B(l+,3,2jl+,l) = B(4,2,l;4,3)

3 1 2 1+ B(3,l,4;l+,2 = B(l,2,4;4,3) B(4,3,l^e) =B(1+,2,1j1+,3)

We can now see that a smaller generating set of is

CB(^3,2;i|,l),B(l|,3,2i3,l),B(^3,2;2,l),B(4,3,l;2,l)}: this will be used in Chapter VI. Recalling the discussion following

Proposition 5.2.2 where the numbers e(H) and 6(H) are

defined, we now prove an important result.

THEOREM 5.2.5 : If H is a c—57? exp-3 loop of nilpotence

class 3 and dimension 4 then e(H) > 2 (and hence &(H) > 3) •

Proof: Let Y y Y2;Yl^ ^6 a bas^s H . We suppose that c(H) < 2 and shall arrive at a contradiction. Since fi(H) > 1 we have e(H) = 1 by our supposition, and every f a € H r has a unique representation as a = gh where g € ft, * 190

ei h e H' XHg is fixed, and 0 < f < 2 . Let A(*<-,3,2) = ± , 6 6 6 A(^,3,l) = 2 > A(^2,l) = g3h 3 , and A(3,2,l) = g^h k

be the representations of the basic 1-associators of H .

We prove the following claim: |{i: ^ 0} | < 2 . The

proof of this claim is also by contradiction: we suppose that

[(i: ei ^ 0] \ > 3 . We may assume w.l.o.g. that e^,e^,e2 € (1,2) .

A(3,2,l) 6 Z(({y^, y2,y^j^J )) implies that for all distinct 6 6 6 r,s e (1,2,3)* e = (B(fh \ v r,Ys) = (h ^,Yr,Vs) “ (h>Yr*Ys) ^ , the second equality following from Proposition 9-2.1 and the fact that c Z(H) • Since ^ 0 we must have

(h,y ,Y ) = e • Thus and similarly (by considering A{^,2,1) I* s and A(if,3,l)) we have

(h,yr ,y u) = e for all distinct r,s € (1,2,3)

(h, yr ,Y s) = e f°r a~l 1 distinct r,s £ (1,2 U) and

(h,Yr,Ys) = e for all distinct r,s (E (1,3,^) .

Hence, for all distinct r,s £ (1*2,3,U) (h,Y ,Y ) = e which * r s by Theorem 2.2.10 forces h (E Z(H) and therefore h ' s Z(H) ,

This is a contradiction to the fact that nil(H) = 3 and so the claim has been established.

By the claim we may assume w.l.o.g that A(3,2,l) =

* h° = £ **2 s Z and A ( ^ 2 A ) - e3 h° = g3 € Hg = Z(H) . Referring to Table 2 and the new generating set mentioned, we 191

see that B(4,3,2;4,l> = B(4,2,1;1±,3) = e, B(U,3,2 ;3,1) =

B(3,2,1;U,3) = e, B(^,3,2;2,l) = B(3,2,l^,2) = e, and

B(4,3,l;2,l) = B(3,2,1;U,1) = e ; but this would mean that

= tej , which is the desired contradiction.

Since a (h ) > 1 , e(H) > 2 implies that 5(H) > 3 * □

COROLLARY 9 .2.6: There does not exist a c-V{ exp-3 loop

G satisfying |g| = 3 ^ and nil(G) > 2 .

Proof: By Proposition 5 .2,2, such a loop satisfies

d{G) = and nil(G) = 3 j but by Theorem 9*2.5 such a loop

would satisfy d(G') = 5(G) > 3 implying that |g'| > 3^

and |g| > 3^ f which is impossible. □

We remark that the Bruck loop satisfies nil(B^) = 3 Q and by Corollary 3.1.12, |b^| = 3 * Furthermore, by

Proposition 3*1*7 we see that has ordered basis

satisfV ine Yi = (xi#x0) for 1 € , A(^,3,2) =

^ ' " ^ 3 + x2,^t- “ *3,^ ' = ^ “^l ^ + *i,U " x3,l^ ' A(*+#2,l) = + Xl, 1* ** ^2 4 ^ J ^(3*2,1) = (0,-x^2 + 3^ - x ^ 3 ) = A(l,3,2)A(if,3,l)_1A(if,2,l) , and

B(lf,3,2jit,l) = B (1^,3,2;3,l)"1 = B(l,3,2;2,l) = B(4,3,l;2,l) =

(0,x^ p , ) . I do not know yet whether there can exist a c—557 exp-3 loop, G, with nil(G) = 3 and |G | = 3^ .

Now all we need is the catalogue of c-/7? exp-3 loops, G , which satisfy |g | = 36 and nil(G) = 2 . 192

PROPOSITION 9 ,2.7 : If G is a c-TTf exp-3 loop of nilpotence

class 2 satisfying |g | =3^ then d(G) = 4 or d(G) = 5 .

Proof: Since nil(G) = 2 we know by Proposition 2.3-12

that d(G) < 5 J and by Theorem 4.3-1, d(G) > 4 since otherwise

we would have , |G .| < 3 4 • □

LEMMA 9 .2.8: If G is a c-??; exp-3 loop of nilpotence

class 2 satisfying |G | =3^ and d(G) = 5 then either

G = R^_ (the loop constructed in Theorem 4-3-10) or

G - y5 - G81 ® i •

Proof: Let G be a c-7?l exp-3 loop of nilpotence class 2 with |G| =3^ and d(G) = 9 - 0 necessarily satisfies

|G'I = 3 ) so again in terras of the discussion following

Definition 5-1.1 we must determine all the equivalence classes of non-trivial ordered 10-tuples of elements from the group

G' = (h^: 0 < q < 2] , We are able to reduce the number of 10 6 10-tuples under consideration from 3 - 1 to 3 as follows.

Since G is not a group, we know by Theorem 5.1.6 that given any basis [y^, Y^jYgj y2,^1^ ® > there exist four elements of this basis which generate = Gg^ ® . So, w.l.o.g. we need only consider ordered bases of the form YgjYg* Y^_) where (Y^Y^Yg) = h and (yJ(.,Y3,Y1 ) = (VVV = (y3,Y2,Y-l) = e . Notice that Y^ has an ordered basis Ty satisfying

Otfry) «= (e,e,e,e,e,e,h,e,e,e) and that II has an ordered basis 193

“ ( Y ^ Y ^ Y y Y ^ Y i ) satisfying a(rp = (h,e,e,e,e,e,e,e,e,h)

(see the proof of Theorem 4.3.10, in particular the case d = 5);

by permuting the elements of this ordered basis we obtain the

ordered basis fR = (y^ = Y^jY^ = Y^Yg - Y£/Y£ = Y ^ Y J_ = YjJ) _ 2 which satisfies = (e>e>h ,e,e,e,h,e,e,e) .

By the above reduction, every non-trivial 10-tuple under

consideration can be abbreviated by the ordered 6-tuple

containing its first six coordinates in order; so we shall

write, for example, o:(F^.) = (e,e,e,e,e,e,H) and a (rR) =

2 — — (e,e,h ,e,e,e,H) where H denotes the common ordered 4-tuple

(h,e,e,e) . We shall eventually show that each of the 3 6 ordered

el e2 e4 e6 - - 6-tuples (h ,h ,h ,h ,h ^ th ,H) is equivalent to afTy)

or to a(rR ) and this will complete the proof of the lemma.

We first show that there are at most fourteen equivalence

classes of these ordered 6-tuples. (We reiterate that by Theorem

4 .2.5 there is a loop G which corresponds to every possible

ordered 6-tuple, and that therefore every ordered 6-tuple

belongs to a well defined equivalence class.) If a loop G

satisfying the hypotheses of this lemma has an ordered basis

T = ( Y^,Y^jYgjY2 ) satisfying the properties of our reduction,

then for 1 < e < 2 and 0 < e. < 2 for i 6 (2,3,4), we consider 5 1 —* the new ordered basis, F'(e^e^e^eg) =

e e e^ 6 ( Y ^ Y ^ Y p = (y^ Y ^ Y - l) 5 (Y3,Yl4.,Y1) 3CY2,Yi+,Y1) = (Yy Y^, Y^ 5

e,5 Si5 8^ (y^,Y3,Y2) = (Yy YyYg) (YyYyYg) = (YyYyYg) h

e e4 e2 e5 (YyY3,Y-J_) * (y5,Y3,Y1) Cy^/Y3,Y1) (Y2,Y3,Y1) = (YyYyY-L) P

e e^ e e • (YyY2,Y{) = (Y5,Y2,Y1) (Yi<_,Y2,Y1) CY3,Y2,Y1 ) J = (Yy Y2>Y^) ?

( Y ^ Y y Y 2 ) = h and (y^ Y ^ y{) = (Yy Y2,y{) = (YyY^Y^) = e .

These equalities are used to obtain the data given in Table 3 below; it shows that there are fourteen types of ordered

6-tuples to consider since ot(r) ~a ( r ' (ey % * ey e2 ^) *

Recall that ct(r) =

( (*Yy Yy Y3 ) f (Ycj) Y[|} Y2 ) f (Yy Y^? Yj_) j ( Y^? Y3> Y2 ) j ( Y^ j Y3$ Y^) t ( Yt^/ Y2> Y^ ) and note that the question of equivalence now boils down to looking at the ordered 3-tuples

((Vy Yy Y^)j (Yy Yy Y-^)* (YyY2> Yj_)) as indicated by the arrows at the bottom of Table 3* 195

Table 3: Equivalence of 6-tuples

a (r"fe5,e^,e3,e2 )) a(f) Type e2 2e3 et — (e,e,e,e,e,e,H) © (h ,h ,e,h ,e,e,H) eP 2e2 ek eR _ (ejejeJe,e,h,H) © (h ,h ,e,h ,e,h ^,H) ®)i ®<=; _ (e,e,e,e,h,e,H) (h jh ,e,h ,h ,e,H)

e? 2e^ % S - © (h^h *,h ?,h 5,H) 2 — ep 2e e, e 2e (e^e,e,e,h,h ,H) © (h ,h ,e,h ,h ,h ,H) ep 2e e e. _ (e^e^h^e^e^e^H) © (h ,h J,h p,h ,e,e,H) e2 2e? S % % - (e,e,h,e,e,h,H) © (h ,h J,h p,h ^e^h ,H) 2 — ep 2e e e, 2er _ (e,e,h,e,e,h ,H) (h ,h ,h -?,h ,e;h J,H)

eQ eK eh er _ ( e ^ h ^ l ^ H ) (h ,h 5,h ^,h ,h ,e,H) e 2e e e, e e,. (e,e,h,e,h,h,H) © (h ,h 3,h 5,h k,h 5,h 5,h) ep 2e e e, e 26,. (e,e,h,e,h,h2,H) @ (h ,h 3,h 5,h \ h 5,h 5,H) 2 — e2 2e^ GtS eU 2Sci — (e,e,h,e,h © (h % h J,h ->,h \ h 5,e,H) 2 — 02 Gci eU 2e5 ep) — (e^e,h,e,h ^h^H) © (h ,h J,h ?,h ,h ?,h ?,H) ®2 2e,R ^ eU 2ec; — (e,e,h,e,h2,h2,H) © (h % h J,h J,h *,h 5,h V ) t t t

We see that ot(rY) is of type (T) and a(rR) is of type

© • Prom now on we shall write (T) ~ (k) to indicate that every ordered 6-tuple of type (T) is equivalent to every ordered

6-tuple of type (k) . 196

Having seen that many equivalences are induced by altering

Ycj , we next consider the effect of altering . If a loop G

satisfies the hypotheses of this lemma and has an ordered basis

T = (y^, Yj^YyYgj Vjl) satisfying the properties of our reduction

and (YyY^Y-^) = (YyYjyYg) = (YyYyYg) = e t then we consider

for 0 < e3*e2 5 2 the new ordered basis of G , r"{e^,e2 ) =

(y£, Yjy Yy Y2j y£) Where y^ = y± for all i € 1^\C4} and e^ y; = Ylf(Y3Jy2 ) . This time we have

e2 (YyY^Yp = (Y5JY^jY3)(y5JY2,Y3) = e .e = e

eo (YyY^Yg) = (Yy Yj^Y2)(Yy Yy Y2) = e *e = e

e3 e2

(YyYyY^) = ( Y^ j Y|^ > Yj_) ( Ycj ; Y 3 j Y^) (YyYgjY^)

(v5",v3",vp - .

(YyYyYp = (YyYyY-^

(YyYyY-p = (Y5,Y2,Y1 )

(Y^YyYg) = (YyYyY2) = h

eg ( Y y Y y Y p = (Y^YyY1 )(Y2,Y3,Y1 ) = e .e = e

e3 (v^ V ^ yJ) *= (Yif,Y2/Y1 )(Y3,Y2,Y1 ) - e • e = e

CYyYyYp = e .

In particular, if a(r) = (e,e,e,e,e,h,H) then a(r"(0,l)) =

(e,e,h,e,e,h,H) and a(r"(0,2)) = (e,e,h2,e,e,h,H); a (r) * 197

(e,e,e,e,h,e,H) then a(r1,(l»0)) = (e,e,h, e^hje^H) and

a( r"(2,0)) = (e,e,h2,e,h,e,H) ; if Qt(f) = (e,e,e, e,h,h, H)

then a(r"(l,0}) = (e,e,h,e,h,h,H) and a(r”(l,l)) =

2 — — 2 — (e,e,h ,e_,h,h,H) j and if ot(r) = (eje,e,e,h,h ,H) then

a(r"(l,0)) = (e,e,h,e,h,h2,H) and a(r"(0,l)) «= (e,ejh2,e,h,h2,H) .

These equalities show respectively that ^7) ~ (2) r (8) ~ (2^) f

® ~ ® , @~®,©~©,@~©,@~@,and

(l^) ~ (^) . So at this point we know that there are at most

six equivalence classes: the classes containing ordered 6-tuples

of types ® , © , © , © , © , and © -

Next we alter . If a loop G satisfies the hypotheses

of this lemma and has an ordered basis f = (y^,y^,y^,yg,y^)

satisfying the properties of our reduction and (y^y^y^) =

(Y^Y^, Y2) = Cv5^Y3,Y2 ) = e , then for 0 < e^,eg < 2 we

consider the new ordered basis of G, r"' (e^;e2 ) =

( Y ^ Y ^ Y j " ^ " ^ ' ) where v"’ = V± for i € I5\(3) and

S2 Y3" = Y^ (Y3Y2 ) * Here we have

Cy^’,Y^t,Y3” ) = (Y5,Y/f,Y3 )(Y5?Ytf,Y2 ) 2 - e .e = e

(Y",,Yi,,,VJP - e

® ’© , y ”') = (Y5,YV Y1)

(Y5'/Y3” ,Y2 ' ) = (y^> Yj^j Y2) \ y5,Y3,Y2 ) - e »e = e 6 6 = (Y j.Y ,,^) 4 (Y 5 ,Y 3 ,Y 1 )(Y 5 ,Y 2 , Y 1 ) 2 198

(Yj^Y^.Yi*') - (y5 ,Y£ ,Y1 )

(Yi",Y",,Y^ ) = (YV Y3,Y2) - h 6 (Y^ f Y^ *^2. ^ = (Y^> Y^j Yj^) ( Yj^j Ygj Yj_) = e * e = e

( Yjli’1 > Yg' t Y^' ) = e

(Y3” ^Yg” ,Yi ) = (Yi|> Yg* Yx) ^(Y3 j Yg? Yx) = e *e = e .

In particular, if o:(r) = (e,e,h,e,e,e,H) then a(rT'r(l7o7) =

(e,e,h,e,h,e,H) ; and if a(r) = (e,e,e,e,e,h,H) then a(r’*'(0,l)) = (e,e,e,e,h,h,H) and a(r',r (0,2)) = (e ,e,e,e,h2,h,H) .

These equalities show respectively that (6) ~ (^9) , (2) ~ (V) ,

and (2) ~ ^5) ; and since © ~ © we know at this point that

there are at most three equivalence classes; the classes

containing ordered 6-tuples of types (T) , (2) , and ^6) .

Finally, we alter Yg ■ If a loop G satisfies the hypotheses of this lemma and has an ordered basis

r - (Y^j Y^, Y^j Yg* Vj) satisfying the properties of our reduction and (y5,Yi^Y3) = (YyYj^Yg) = (YyY^Yg) = (Y^Y^Y^ = e, then we consider the new ordered basis of G , r,Mr =

(Y5%y/;,’,Y',',,Y^'jVi”) where y”" = y. for i 6 1^ ( 2 ] and

Yg" - Yj^Yg * Here we have

(Y’,”,Y;,,,Y^") = e

( y'"',y; ' ,,Y^”') - (Y5,Y^,YS) - e

( y£ " , yJ",y£") = (Yj.Y^Yi) 199

(Y’’^Y ’M,,Yr) = (Y5,Y3,Y^)(Y5,Y3,Y2) = e *e ^ e

(Y^Y^Yp = e

(y, " " ^ " ® " ) = (Y5,YJ+,Y1)(Yy Y2,Y1)

CYiM,,Y'”',Y2") = (YV Y3,Y2 ) = h

( Y ^ Y ^ Y ^ ’) = e

( Y ^ Y ^ Y p = (y^ Y ^ ) = e

(y® Y 2" , Y ® = (Y3,Y4,Y1)(Y3,Y2,Y1 )

So in particular, if a(r) = (e,e,h,e,e,e,H) then a (r"") =

(e,e,h,e,e,h,H) which shows that ((T) ® . Since ® ©

we have (^2) ~ (&) . Hence, the only two equivalence classes

are the classes containing ordered 6-tuples of type (T) which

is the class of a f r ^ ) and the class containing ordered 6-tuples

of type ( ^ ) which is the class of a ( r R ). □

LEMMA 9 -2*9: If G is a c4j exp-3 loop of nilpotence class

2 satisfying |Gj = 3 and d(G) = 4 then G = K (the loop

constructed in Theorem 4.3*13)•

Proof: Let G be a c-77J exp-3 loop of nilpotence class 2 with |G| =3^ and d(G) = 4 . G necessarily satisfies 2 |G’j » 3 , so we must show that there is a unique equivalence class of non-trivial ordered 4-tuples of elements of

^ ql Gf *= (hp h. : 0 < 9-j^Qp < 2) • Furthermore, w.l.o.g. we need

**2 qi ^ ql only consider ordered 4-tuples of the form (h^h^hg h1 h ) 200

since we can assume w.l.o.g. that any ordered basis r =

(Yjy Yy Yg, vi) of G satisfies (Y^YyYg) - and

(Yy Yy Yj_) ~ hg * (Note that the ordered basis fK constructed for K in the proof of Theorem 4,3*13 satisfies Qi(rv) =

(h^h^e^e) .) Since 0 < < 2 > there are 3^ such ordered 4-tuples to consider.

We first alter y^_ * If G is a loop satisfying the hypotheses of this lemma and has an ordered basis r = (YyYyYg, Y^_) which satisfies (YyYyYg) = ^ and (YyYy Y^ = hg then we consider for 0 < ey e3 5 2 the new ordered basis of G, r T e ^ T ^ T = (YjyYy Yg,Yj_) where y^ = Y± for i £ I^\(l) and e4 °3 Y1 = y4 (y3 vi) * We have

(YyY^Y^) = hx

( Y ^ Y y Y p = (yv y3,y-l) = ^ 0 2e 3 3

e4 e4 (YyYgjYj^) “ ( Yy Yg, Y^ ) (YyYg,Y-j^) — (YyYg,Y^) *

These equations provide the data in Table 4 below which shows that there are only nine types of ordered 4-tuples to consider. 201

Table : Equivalence of li--tuples

a(r'(e4,e3 )) a(7) Type 2e^ eh (h-^hg^e) © (h^>hg,h^ >h^ ) 2e e, (h-^hg^hg) © (h^hg,!^ tbghj4 )

2 2e^ 2 elj. (h1,hg,h1 ,hgh^ ) ^hl'h2,e,h2^ © 2e e, (h^ h^ h ^ e ) © (h1,hg,h2h1 2e e, (h^, hg# hg^ hg) © (hp# hg,hghi jhgh^ )

2 2e^ 2. elj- {hj^# hg j hgy hg) © (h1,hgJh2h1 * , 1 ^ ) 2 2e e. (h^j hgjhg# e) © (h^,hg fhgh^ fh^ )

2 2 2e^ elx. C h^ j hg j hg f hg) ® (h^hg^hghi jS,hgh1 )

P 2e^ 2 ei± ( h ^ h ^ h ^ ) © ( h ^ h g ^ h ^

Our final step is to alter Yg * Tf r = Cy^jY^jYgjY^) «(r) = (h^h^e^e) then for 0 < e^,e^ < 2 we consider the new ordered basis of G, r"(e^,e^) = (y£j Y^#Yg*Y^) where e, vi = Yi f°r 1 € and Y2 “ Y^ (y3 Y2) * We have

(Y^Y^Yg) = (YU,Y3,Y2 ) = \

(VvVyYi) = h,

(y£,Y2,yJ) = (y^jY^jY^) (Ylf,Y2,Y1) = .e = hg-3 202

e 2e, Hence, a(rr,T(e^,e^)) = (h^h^hg ,h2 ) and in the notation

of Lemma 5 .2.8, (T) ~ (k) for all k £ 1 . So, the equivalence

class containing (h-^hgjeje) = a(rK) is the unique equivalence

class of ordered 1-tuples and therefore K is the unique c-^f

exp-3 loop of nilpotence class 2, cardinality 3^ , and

dimension 4 . [?

The following theorem is an immediate consequence of

Corollary 5 .2.6, Proposition 5 .2.7, Lemma 5 .2.8, and Lemma 5 .2.9*

THEOREM 5 .2.10; There are exactly three non-associative 6 c-7?{ exp-3 loops with cardinality 3 : R^, K, and .

Recall that |Z(B^) | =3, |Z(K) | = 32, |Z(Y^ ) | = £ , d(l^) = d(Y^) = 5, d(K) =4, |B^ | = |Y^| = 3, and |K’ | = 32 * charter VI

COMMUTATIVE MOUFANG EXPONENT 3 LOOPS

OF NILPOTENCE CLASS 3

§ 1. The Behavior of the Product

In this section we determine the multiplicative structure

of finitely generated c-V{ exp-3 loops of nilpotence class 3 using the method of § 1 of Chapter IV where the product of

finitely generated c-771 exp-3 loops of nilpotence class 2 was analyzed. We take particular notice of the case of such loops having dimension The situation here is more complex than before and to make the exposition more palatable we shall use simpler notation which is described in the following paragraph and the succeeding definitions.

Throughout this section we assume that G is a finitely generated c-ty exp-3 loop of nilpotence class 3* We again use

Lemma 3*1*13 which allows us to represent an element y 6 G

yd yd - 1 y2 yl uniquely as y = hytVd Cvd_1 (**«(y2 )•*»))] where f = (Yd> Yd _ lt • • 'jYgj Yd) is an ordered basis of G, 0 < yt < 2 for all t £ I , f and h € G T • Furthermore, recalling that u y (G',G',G) c c (e) implies that G* is a group, we can write y h = g • TT A(t,i,j) where g, € GP and as in y y l

203 20k

Lefinition 5 .2.3, A(t,i,j) = (y,,YjjY*) * The representation of "C 1 j h may not be unique if there are dependencies among the J elements A(t, i,j) ; but for each element h € G' there is y always some such representation, so we choose one and fix it

throughout our discussion. We shall again be concerned with the

multiplication of two elements of G : z z z z 2 - (gz * TT A(t,i,j) ^ ^ H y/ C . - . ^ Y i1)"-)] and 1 < j < i < t < d

y * y y y y = (k^ • TT A(t,i,j) ^^^[^/(...{Yp^l)...)] . Many y l

of the terms in the following definition are the same terms

defined in Definitions 4.1.1 and 4.1.2, allowing only for our

change in notation.

DEFINITION 6.1.1: For z,y 6 G having the above

^t ^t — 1 zp ^1 representations, we define TT = \ (y (...(y y )...)), z t X> t w “ 1 d 1

\.t " *z.y(M) = Vj - Vi'

Cz.yW - st - yt’ ez.y " - yt,i,J’ fz.y(t) "

*t + yt. fz.yCt,i.J) - FZjy(t) -

t. T(t) f v(t-l) f. y(2) ^ „(l) YtZ'y ( Jy (...(Y/'y YxZ'y )...)), (t.i,5) = c z )' an<^ X) *= cz^^(t,i, £)

When there can be no ambiguity, that is when we are involved with just the product z ■ y , we shall use the following 206

abbreviations: M 1 l .1 L ft' and ^(t.i,j;k. £) .

DEFINITION 6.1.2: For z,y € G having the above

representations, we define

^ -y(t,l,j,k) = Cz,y(t> < W k * W i ’

^ y(t,J,k;i,k) + 17^ (1,3,K}t,In) -

d,k) + (z±\ + y±yk )^^yCt,d,k) -

Q^y(*,i,j^k) = 2 ^ y(t,i,J;J,k) - ^ y(t,d,k;i,j) + ^ y(i, j,k;t, j) +

7 ^ y(t,i,k,o) + (ziZj + y . y . ) ^ y (t,j,k) -

fz,yCt)fZ,y(J)^ , y (i'J'k)

- ^ y(t>ijk;i,J) + J^^y(i,j,k;t,i) -

(zi + y±)\,y (t,j,k) - fz>y(t)fz>y( D ^ y(i,d,k)

Qg y (t,l,j,k) - S^,^y(t,i,J;Mk) - !!^>y(t,i,k;t,d) + » y(t,j,k;t,i)

DEFINITION 6.1.3: For z and y having the above representations and for all t > 3 we define S? (t) = c (t)------fYt ^ >\,t -ljTy,t - 1^ ' * a . y ^ =

t - 1 t - 1 f (i) f (t) ( TT ft (k) , TT y.Z,y , Y.|.Z, y ) (note that because the k = 3 'y i = 1 1 % left argument is an element of G1 - Gnil(G) -2 = Df°> see Definition 2.2,15, "the middle argument need not be bracketed)

where an empty product of elements of G is defined to be e ,

We see in the following lemma that the elements ft (t). z,y " X- and & (t) are those that arise in the multiplication

of the elements z and y .

LEMMA. 6.1.U: If G isac-#; exp-3 loop of nilpotence

class 3 with ordered basis r = i> and

z,y € G have the above representations z - W d and y = d then z *y = (hzhyv3(z,y))Fd where v3(z,y) =

d d

Proof: We first prove the following claim: for all t' with 3 < t ' < d , 71

The proof of the claim and the proof of the lemma itself involve many uses of the facts that G' £ D(G), (G',G',G) = (e), and

G2 c 2(G) as well as the properties of associators given in

Theorem 2,2.1 , The claim is proved by induction on t ' } the first case being the case of t ’ « 3 . [ ^ Y3 ' V 2'Y3* 3 3^Y33jTV,2J^ Y3jTTz,2'Ty, 2 ^ Try,2/Y3J 3 3] “ Zo CY3 jTT2 2^ by ^associativity and Theorem 2.2.1 . Hence z ^

^ " y . s = (y33’V ' V )[v33(\ , 3 V )] ■ y y y = ^Y33rV,2)Trz,2 = [ Y33^TTy,2TTz,2)] (y3 ^ V 2*^,2 >

and since (V3> TT^2, TT^ 2 ) € Z (<( Yy ^ 2, T^ 2 ) >) we see that

Y33^TTy,3Tz,2 ^ = [ Y33^Y33[TTy,2TTzJ2 ^ ]^Y33,TyJ2#TTzJ2^ = f, -y. [V, (TT 0TT „)](y, ,TT o>TT ) . Combining this with equation (1), j zj<= y#^ j y,^

*e see that 7^ 3^ 3 =

^ ’V ’V ^ ^ ' V ' V ^ ^ V V ” =

Rz>y(3)[V33(IT^2Try^ )] = Hz>y(3)(Yj3F2 ) = RZjy(3)F3 =

3 3 ( TT y (t) * TT ft (3))F_ t with the third equality following t »Jf Z>y t =3 Z,y J from diassociativity: TTz 2>TTy 2 € ((y2'y1^ * So, the claim has been established for the case of t' = 3 *

We now assume that d > t 1 > 3 and that the claim has been established for t ' - 1 . Exactly as in the above computation for the case of t ' = 3 , we calculate that TT . , • TT . , = z, t y,t

ft ’ Rl ..(t'JtYa., (TT . , n TT , )] . Then by induction we have u zjTt ■ l “ X V f ' V ' = ^ y (t’)[Yv ’V u \ y(t) ■ J 3\ y (t)lFt'

= (t ! i N * y(t) ' ^ y (t’)[V ,(J 3\ y (t) ,Ft

since V (t) fG, c z(G) for all t with h < t < t' -1 . z,y — —

Now,

f t * - l

Yt*’( "C = " j Bz,y(t) ‘Ff -1 > =

f t * — 1 t'-l f [fV Ft'-lJ *t^3 Rz,yft^ t 53 Rz,y(t^ Ft'-l ,Yt* * =

t'-l (F., • TT ft (t))Y (tf) v t’ t =3 z,y " Az,yv '

t'- 1 since TT ft (t) € G* = D{G) . t = 3 Zjy Hence,

t' t'-l t, *TT =( TT ?T (t) -ft (t*))(F • TT ft (t)) y>^ + _k z jj z>y **- t = 3 z>y

t ' t * = ( ffVz v (t)- IT R (t))F . t =!f 'y t = 3 ,y 11 and the proof of the claim is complete.

Now, for the proof of the lemma; z *y = (h TT ,)(h TT ,) z z,d/Y y y,d'

Wehaye C-z.^d'Vy.d)

(hz’ TTz,d 'hy ^ hz 'IIz,d ',Ty,d) = ^hz’ ^Zjd' ^ d ^ and

K ^.(h TT A) = [h (TT .TT .)](h ,TT ..TT .) . zjd y y,d/ y y,d z,d/J' y> y,d* z,dy

Thus, Finally, by the claim we see that

d d

= (hz V 3cz>y))Fd •

For the remainder of this section we shall assume that d(G) = ^ and that r = ( Yj_) is &n ordered basis of G note that in this case v„(z,y) = (^)R (3)ft (*+) ■ 3 * z,r z,y z,y z,y We shall abbreviate Q/f (^,3,2,1) by Q for £ € Xj, J we ?y * ^ begin to see the importance of these functions in the following lemma.

LEMMA 6.l.g: )ft^y(3)«2^y( ^ ) =

TT A(t,i,J)*l(t'i'J)B(U,3,lS2,l)^IB(^3,2;2,l)(^ 1< j < i

B(^3,2;3,l) JB(lf,3,2;lf,l) ^ .

Proof: The proof is a lengthy computation using the expansion law repeatedly. We recall from Table 2 in §2 of

Chapter V that * OJ N > s --- - r 4 H I HH /S ** 1 H H H H OJ •l • 4 ■\ •X ** H CO m o n •i m • * ■ A ^ — OJ C\J CM CM CM < • 4 *v •s •k o n n o n o m n o 0 ) ii ** *\ * *x u < " - = f - = t 0 J w Ii ' d £ i C 1 w CQ PQ n > OJ OJ 0 3 N > » . 11 u 11 ii H ,— , . H N n o OJ > ■ n o CJ n o OJ o * OJ ♦x ** «"“ N i - i OJ > m - d ~ CMH > •% * * *«* *A *#\ V w J p •x *t f-| H H H H o n > - •* * t = * > •t CM n o OJ n o *\ (U •* •s •% *x o j OJ > n o - r h •< 3 11 II " w * ' ' w ' . n r H p q CQ PQ PQ II t = > OJ H •s N > > , H 0 0 H N > ■ ii , H N > » H O J O J H > N > i H II OJ OJ 1-— . > H H *\ > > O J r-t OJ I 1 --— . y^N > »s OJ 1 = •l OJ > H H H ■ a H •» * 1 OJ •V •* •k •x •V v - ’> 1 1— 1 > - H H CM f c l CM CM CM M H •V •« > ■ #s *#* *«k • i\ •i > OJ H H CM CM OJ OJ > ■ •i o n •x N O J •W o n > n o•sn o•v n o n o . tsi > >— ■■ v__ / *> *\ II m J* II ■'w' *d“ •\ o n o n > OJ ii H r q PQ (X) PQ 0 m N t J < ^ S n 1J 11 11 ll OJ H 0 1 •V ^*v z 1 •» II w > > > £»i H H CJ CM t = •k OJ •X •x *x *i *» OJ •\ n o -M- -=t c o o n OJ N ^ O J H ** ■ Ik • *\ S w f f c : > > > - H H H H •> * 1 •i *\ * •x •v *» OJ ^ o n H o n CM CM CM CM •* > ? > *\ *x •x •x l M ' n o n o t = ** It II li m ffl ffl PQ o n H H OJ 5 E h s 211 y2 y x , (yv Y3,TI^2 ) = (y4>Y3,Y2 Yx ) = (W6W? )(t6T7) (10) where

y y

W6 85 (V y3*y2 J 2 = a C^3,2) 2 (11) y2y 2 \ 2*1 y0y T6 =

y y W7 = (vV Y3,Yl^ 1 = A(^3,l) 1 (13) and 2 2 y y H y y T? = (a(U,3,i),y1,y2 ) 2 1 = B(U,3,1;2,1) 2 1 (Ik) .

Prom equations (8), (10), (11), (12), (13)# and (ll+) we obtain

y y y 2 y « y y 2 W5 = [A(l|,3,2) A(lf,3,l) 1][B(U,3,2;2Jl) 2 h(k,3,l^A) 2 ^ (15)

Prom equations (3), (7), (15), and (9) we obtain

z . y 2 z y W3 - [A{^,3,2) J A(i+,3,1) J n 2 2 [B(J+,3,2;2,1) j **Z?y2yi J 1] (l6).

Next, z.y2 z,y, z z •3 A Hi ? 1 1 3 1.. .2., ±\ CV Y3,TTz ,2 ) = fA<^3,2) J A(U,3,1) J Yi>

Z^!Ziy2 Z^iZ?yi = B(i+,3,2;3#1) J 1 B(4,3#l;3,2) J 2 1

-z A(2,l) = B(4,3,2;3#l) J (IT), so from equations (5) and (17) we obtain

-z2M2,1) T3 = B(U,3,2;3,l) J (18) .

Z 2 ^Y4 ,Tz,2 ,Ty , 3 J " (\>y2 \ > ^ , 3 ^ ° fW8W 9 J(T8T9 J (19j where 212 2 2 Wg = (\>V2> \ t3) 2 (£0) «9 ' ( W j / T T ^ ) 1 (2D 2 2 Z Z t8 = (w8'Y2'Yl> 2 1 (22> and T9 = C V Y1,Y2 ) 2 1 (23) * y. fv^jYgjTT^^) — (v^jYgjY^ TTy g) = ^ X O ^ H ^ ^ 1 0 ^ 1 1 ^ (*^) where y —y W10 = < V Y2'Y3 ) 3 = A(U,3,2) 3 (25) = (yU,v2 ,TT ) (26)

y y T10 = (W10'Y3',V ,2) ^ and T11 ■ (WIl',ry,2'Y3) 3 (28> •

From equations (27) and (25) we obtain 2 1 2 y y -y —y y T10 = (a(^3,2),y3#y22Y11) 3 = B(4,3,2;3,l) 3 1 (29) and

from equation (26) we obtain

ya y-, y- wn = (yif.Y2,Y22Y11 ) = a(U,2,i) 1 (30).

y y y (Wll'1Ty,2'Y3> = (A(‘‘>2»1)»V2 YX ,Y3) 1 = 2 2 y2yi y-i y?yn yn B(4,2,l;2,3) Ti(M,l;l,3) X = B(4,3,2;2,l) ^ ( U ^ l ^ l ) 1 (31) and from equations (28) and (31) we obtain y^ypy-, y^y,2 TU = B(^3,2;2,l) 3 3 (32) .

Now, from equations (20), (24), (25), (30), (29), and (32) we obtain 2 ~zny-x "Zpy^yi Wg = A(4,3,2) d A(4,2,l) 2 n(4,3,2;3,l) 2 3 1 zpy7ypy, zpy^yi2 B(4,3,2;2,l) 2 3 d TJ(4,3j1}2,1) 2 3 1 (33) . From equations (22) and (33) we obtain

“Z 2 2 y 22 I 8 - W *,3,z),ys,v1) 2 1 3(A(i*,2,i),v2,Yl) 2

-zzy B(4,3,2;2,l) d -1 3 (3^) .

Hext, (Y^Y^IT ) = (Y^Y^Y^TT „) = (W^W )(T T ) (35) y,3 '»4'Tl'r3 y,2 ' ~ v 12 13 M 12 13

y- -y^ where = ( Y ^ Y ^ Y g ) = A(J+,3,1) (36)

W13 = (YU ' V TTyJ2 ) <37> T12 = (Wl2 'Y3'ny,2 )y3 (38) and

y-, T13 = ^W13,1Ty 2,y3 ^ ^3^) * From equations (3 8 ) and (3 6 ) we obtain 2 2 y2 y-1 -y, -yqyP = (A(if,3,l),Y3 ,Y2 y± ) = B(U,3,1;3,2) 3 =

2 y^Yo B(4,3,2;3j1) (^°) Edid from equation (3 7 ) we obtain y y —y W13 = < W y22y11) = A f M * 1 ) 2 C*H) •

y y ,*y (W13, y 2 ,Y3 ) = (A(I»,2,1),y 2 Y11,Y3) = 2 -yP -Yo Yy B(U,2,1;2,3) B(^,2,lil,3) = 2 -yp -ypYn B(^,3,2;2,l) B(4,3,l;2,l) ^ 1 (t2 ) and from equations (3 9 ) and (^2 ) we obtain Now, fjrom equations (21), (35), (36), (^1), (^0), and

(^3) we obtain

-z y -z y Wg = A(if,3,l) 3A(lf,2,l) 2 2 -s-.y-.yp -znypyPj B(4,3,2;3,l) 1 3 B(^3,2;2,l) 1 3 ^B(U,3,l;2,l) 1 3 ^

From equations (23) and (J+if) we obtain 2 2 ••Zf 2t y «z z t9 = (A(^,3,1),Y1,V2) 2 1 3(A(V,i),Yl,Y2) 2 1

B(M,1;2,1) Z2Zly^ 3 (^5).

From equations (4), (19), (33), (^), (3*0, and (**5) we obtain

= A(it-,3,2) ^ ( ^ 3 , 1 ) iy3A(U,2,l)A(2^> 2 , „2 Zpy^yn -^-.ypypy-, +zpziyp b(4,3,i;2,i) 2 3 1 1 3 ^ 1 2 1 3 2 2 Zpy^ypYT - ziypyp - s zi ^ B(^,3,2;2,l) 3 ^ 1 1 3 ^ 2 1 3 2 2 -zPy^yn + z-ty^yp B(M,2;3,1) 3 3 (^6) .

From equations (6) and (^6) we obtain Finally, from equations (1), (2), (16), (46), (18), and

(47) we obtain

Cl (2^0 "* zpy-3 ) Cli^Z'3yi ® .A(2,l) y (^) - A(4,3,2) 3 2 23 A(4,3,l) A(4,2,l) '

B(4,3,lj2,l)C^^"Z3y2yi + " Zly3y2yl + Z2Zly3 + Z3Z2zlyl “ Z3Z:y2 )

/ 2 2 2 2 % (z^ypy, + zpyPyPy, - z-,yPyP - zPz-,yq + z^z^yi - z?z~z^ B(4,3,2;2,l) 4 3 2 1 2 3 2 1 1 3 2 2 1 3 3 2 1 3 * 1

B(^3,2;3,l)0'(('Z2y'yi + Ziy^ ' Z34(2'1>)

\ ^Z3Z1 + y3yl ^ ^ ,2,'L^ v (4j3>^j1) b (4,3,1;2,1) ,y

(Z-3Z2 + y^yp vf^>3>lj2) B(4,3,2;2,l) 3 ^ ^

■(z? + y? M M , i ) B(4,3,2i3,l) 3 3

Next, we compute y(4) and ^(4); the computations are not as lengthy.

f f, f. = (A(3,2,l)*i(3,2#1) , V33Y22Y11 , )

V ^ 3'2'1J V p^ 3*2*1) - B(3,2,l;3,4) 4 3 B(3,2,l;2,4) 4 ^ 216

= B(^,3,2i3,l) 5 BC^3,2;2,l)

-f. f\ M 3 , 2 A ) b (U,3,i ;2,D

Jr (k) - (h h"1 . TT ,, . TT ) z,y ' v z y * z,4 J

= ( tt A(t,i,j)Ct^ ^ , rr Y*k, rr v^) 1

rr (A(t,i,j),y 1< j < i < t <4 K 1 (k,x) e I2 k^ t Ct ± ^(k,x) TT B(t,i,j;k,je) Z>X>3 1< j < i< t<4 i< je

B(^3,l;2i1)S^ ^ 3,1S2,1^B(l|-,3,l;3,2)^l,‘l3'lj3'2 ^

B(^,3,l;^2)^,3'1;^ 2^B(U13,2;2,l)^^3»2i2'1J

B(^,3,2;3,l)^^,3,2;3,1^B(if,3,2;4,l);!?r^,3,2slf,;L^

= B(iT,3,l;2,l)^'3,lj2,l) ’^ 2^1J3'1) +^3,2,1;U,1)

B(U,3#2;2,i)^^*3*2;2#1) -2?f(lfj2#l;3j2) + 5?|(3j2,l;l4-,2) 217

B(K,3,2;3,lfl(k,3,2i3*1) - ^ 3'lj3'2> +^C3,2,l;lt,3)

Hence, « * A(t,i,J> .

□

DEFINITION 6.1.6: If G is a c-ty exp-3 loop of nilpotence

class 3 with ordered basis f = (yd,...,y^) then we define _ d Xo(G,T) to be the set G' x X <[y. .}) endowed with the -*------i =1 " 1

zd Z1 yd yl binary operation * defined by (hz,vd ,...,y1 )*(hy,Yd ,...,y1 ) =

fd fl (hzhyV3(z,y),Yd ,...,y1 ) where v^(z,y) =

K v(d) * " *z v(t) * * Rz v (t) * z'y t = b >y t = 3 z'y

We can now give an analogue of Theorem U.1.7 •

THEOREM 6.1.7: If G is a c-771 exp-3 loop of nilpotence class 3 with ordered basis r = then G = x^fG^r) *

Proof: By Lemma 6.1.U- the bisection ifr: G ■* x^C^T)

Zd Z1 z (hz,yd ) given by Lemma 3*1*13 is an isomorphism. 2l8

COROLLARY 6.1.8: If G is a c-7?? exp-3 loop of nilpotence

class 3 and dimension 4 with ordered basis r = Yy Y2> Y-^)

and if each element hz € GT has the fixed representation z+ -t -i h = g • IT A(t,i,J) * where g € Go and Z Z l

0 < z . < 2, then G = xq(^r) where v,(z,y) «

ir A(t, i, j 1< j < i < t <4

Qi Qo Qo Q(, B(4,3,l;2,l) B(4,3,2;2,l) B(4,3,2;3,l) JB(4,3,2;4,1) ^ .

Proof: The proof is immediate by virtue of Theorem 6.1.7 and Lemma 6.1.5 « □

§2. The Construction of F„ . ______

DEFINITION 5.2.1: For d > if we denote the free c-37? exp-3 loop of nilpotence class 3 and dimension d by F_ , .

Note that by Theorem 2.2.4, ^ is the free c-ty exp-3 loop of dimension 4. So, the HTS, S(F^ r ) , corresponding to

7 . is the largest HTS (X, fi) satisfying dim(X) = 4 in the associated linear incidence structure (X, fi,6) . Before constructing F- r we show that any c-5exp-3 loop, H , satisfying 8 nil(H) = 3 and d(H) = 4 must also satisfy |n| > 3 . (Again, we recall that the loop satisfies nilfB^) = 3 , d(B^) = 4 , 219

g and |B^ ] - 3 •) By Theorem 5.2.5 such a loop H necessarily

satisfies |h| > 3^ •

PROPOSITION 6.2.2; There does not exist a c-ty exp-3 loop

H, satisfying d(H) = h, nil(H) = 3, and |h| = 3^ *

Proof: Suppose H were such a loop and let T = (y^, y^, Yg,V^)

be an ordered basis of H ; we shall arrive at a contradiction.

By Theorem 5.2.5 we must have e(H) = 2 where H' = Hg ® ^3^^ *

We must also have |Hg | = 3 ; with Hg = <(g) > we consider the

basis [g,h^,hg} of H ' . S G S 1 S G G * So, when we write A{4,3,2) = g > A( ^ 3 A ) = & *

s e e’ s, e, e/ A(4,2,l) = g h2 , and A(3,2,l) = g h1 hg , we may assume w.l.o.g. that e^ = e2 = 0 and that e^ = e^ = 1 ; that is,

s2 we may write A(U,3,2) = g^Ti and A(lj-,3,1) = g hg . Hence, since g € Hg ^ Z(H) , we see that ^ € Z(<{y^j YyYg) >) and lv> € zC<(Yu,Y3,y1}» .

Now, B(4,3,2;lf,l) = = B(U,2,1;4,3) J and on the other hand since Hg £ D(H) n Z(H) ,

s i _ B(^,3,2;i|,l) = (g 'T11,Y^,Y1) = ( h ^ y ^ y ^ ,

-1 S2 -1 -1 B(lf,3,li^,2) = (g h2jYif>Y2 ) = O^Yj^Yg) > and g 0 0* 0 01 B(4,2,l;4,3) = (g 3h13hg3,yIt,Y3) = (h^Y^Yg) 3 = e.

Hence, we know that ( h ^ y ^ y ^ = (h1,Yif, Y3) *= (h-^y^Yg) = 220

® and (^gjY^jYg) “ f^gjY^^Y^) = ( ^ jYi^jY^) ~ = e .

Next, B(M,2;3,1) = B ^ ^ l t f ^)"1 = B(3,2,l;^,3) J and on the other hand, B(^,3,2;3,i) = (h^Y^Y ^! B(U,3,1;3,2)-1 =

Gt G * (h2^Y3,Y2 )"1, and B(3,2,l;4,3) = ( h ^ Y i ^ ) ( h ^ Y ; ^ ) ^ = e .

Hence, (h1jY3#Y1 ) - e and (h^YyYg) = e •

Finally we have ( h ^ g ^ ) = B(4,3,2j2,1) = B(3,2,l;^,2) = ek ei| (h^^Y^jY2 ) (^2■* Yj|# Y2) = a and (^jjYgjY^) = B(^,3jl»2,l) = e, e/ B(3,2,l;l*,l) = (Il^Y^Y-l) (h2jYi1.#Y1) = e .

So by Theorem 2.2.10 we would have € Z(H) , implying

that H' c Z(h ) which is a contradiction to nil(H) = 3 • Hence,

no such H can exist. □

We remark that the proof of Proposition 6.2.2 actually

gives the following stronger result: if H is a cexp-3 loop of nilpotence class 3 and dimension U then e(H) > 3 .

There is still the possibility that there exists a c-57f exp-3 loop, G, satisfying |g| - $ f nil(G) = 3, and d(G) = 5 . The investigation of such a possibility must be postponed until a study has been made of the general structure of c-ty exp-3 loops of nilpotence class 3 and dimension 5. Such a study will entail an analogue of the analysis which led to Table 2 in §2 of

Chapter V; the key tools in this analysis will be the five element 221

identities given in Proposition 2.2.18 and the five element

identity to he given in Proposition 6.2.3 ■ (See [51].)

We have seen in Theorem 5.2.5 and Proposition 6.2.2 that

we do not have the same freedom in constructing c-771 exp-3 loops

of nilpotence class 3 that we had in constructing such loops

of nilpotence class 2. The reason for this is that relations

among elements of the associator subloop induce other relations;

for example, if A(4,3,2) = A(4,3,l) then B(^,3,2j3,l) =

B(4,3,l;3,l) = e . However, in the case of F_ , there are no

such relations and we are able to construct F_ . "freely."

We have deduced from the data of Table 2 in § 2 of Chapter V

that if T = (y^, Vgj Yj_) an ordered basis of F^ ^ then

F^ ^ is generated by {A(t,i,j): l

{B(4,3,2;tf,l),B(U,3,2;3,i),B(^,3,2;2,l),B(U,3,l;2,l)J . Hence, 22 we know that |F_ < 3 ; and when we construct a loop of 1 p cardinality 3 } nilpotence class 3> and dimension k we know

that it must be isomorphic to F_ j .

X have very recently learned through a private correspondence

flam Jonathan Hall that Marshall Hall in [ 31] has found a set

of four independent elements of the Bruck loop ft which generate

a subloop of cardinality 3^ which has to be isomorphic to

F„ . . Once he has been clever enough to find the right four

elements, his representation of F . turns out to be a good bit

simpler them the one I am about to present. However, my method

has the advantage that it can be applied (see [51]) in conjunction with the appropriate generalization of Lemma 6.1.5 , the five 222

element Identities of Proposition 2*2.18, and a farther five

element identity (to be given in Proposition 6.2.3) for c-V{

exp-3 loops of nilpotence class 3 to construct F.

d > 5 . The only other construction of an infinite family of

non-isomorphic o.-7f\ exp-3 loops of nilpotence class 3 was given

by Bruck in [5]* For d > 4 he uses the Burnside group

B(3,d-1) (see [2^]; B(3,3) was described in the final

paragraph of §2 of Chapter I) to construct a c-ty exp-3 loop

BUj satisfying d(BU^) = d , nil(BUd ) = 3 j and IBU^ | =

It will be shown in [51] that |F_ .|

PROPOSITION 6.2*3: If G is a c-^ loop of nilpotence

class 3 and if v, w, x, y, z are elements of G then

((v,w,y),x,z)((v,w,z),x,y)((v,x,y),w,z)((v,x,z),w,y) « e .

Proof: Let a = (v,wx,yz) and let p » (<2,vx,yz) . On one hand p = e since a 6 Z(({v,wx,yz))) .

Now by the expansion law, a = (v,w,yz)(v,x,yz)g' where g' € G2 = Z(G) . So a = (v,w,y)(v,w,z)(v,x,y)(v,x,z)g where g € Gg £ Z(G) . Therefore, since G 1 c D(G) we have

P * ((v,w,y),w,y)((v,w,z),w,y)((v,x,y),w,y)((v,x,z),w,y)

((v,w,y),w,z)((v,v,z),w,z)((v,x,y),w,z)((v,x,z),w,z)

((v,w,y),x,y)((v,w,z),x,y)((v,x,y),x,y)((v,x,z),x,y)

((v,w,y),x,z)((v,w,z),x,z)((v,x,y),x,z)((v,x,z),x,z) . 223

The first, sixth, eleventh, and sixteenth associators in this

product are equal to e . Hence,

0 “ ((v,w,z),w,y)((v,w,y),w,z)((v,x,y),w,y)((v,w,y),x,y)

((v,x,z),w,z)((v,w,z),x,z)((v,x,z),x,y)((v,x,y),x,z)

C(v,x,z),w,y)((v,x,y),w,z)((v,w,z),x,y)((v,w,y),x,z)

= [ (Cz,v,w),w,y)((y,v,w),w,z)][ ((x,v,y),y,w)( (w,v,y),y,x) ]

[ ((x,v,z),z,w)((w,v,z),z,x)][ ((z,v,x),x,y)((y,v,x),x,z)]

[ ((v,w,y),x,z)((v,w,z),x,y)((v,x,y),w,z)((v,x,z),w,y)] .

Now by part (ii) of Proposition 2.2.11, each of the first four pairs of associators above consists of an element multiplied by its inverse. Hence, e = 3 *=

((v,w,y),x,z)(fv,w,z),x,y)((v,x,y),w,z)((v,x,z),w,y) . □

DEFINITION 6.2.4; We let S_ denote the set GF(3)^ and we write an element z G S as

z - tzi'z3'22,zr Z3,2,i'zif,2,l'zif,3,ljZi+,3,2,V z3,V zl ] * We define a binary operation * , which will simply be denoted by juxtaposition, on S coordinatewise according to the following rules:

^zyh = zx + y t for 1 € h

“ \, 1 , 1 * + ^y^M) fOT

m = zm + y; + ror " € \ • 2 2 1 ;

Note that the functions 77L (t,i,j) involve the elements 2.>y z ^ and y^ for £ € 1^ and the functions y (^>3i2,l)

involve the elements and for £

z. . . and y for l

twist."

We have seen in the proof of Lemma l;.2,10 that the functions

7ftz,y , (tjijj) satisfy a functional ^identity "formally"; that is. + s\jZy(t,i,3) + ^ x(zy)(t,i,d) -\ Z(* A ,3 ) *

+ \z , x y ^ i,J^ ' ThiS WaS a key P°int in the construction of c-ty exp-3 loops of nilpotence class 2, and the

corresponding key point for the case of nilpotence class 3 and

dimension 1; is that the functions Qm (t,iltj,k) also "formally" zt y satisfy this ftmctional ^identity.

The following theorem will be proved by the seven lemmas

which succeed it. It is a partial converse of Corollary 6.1.8 .

THEOREM 6.2, 5 : (S, *) is a c-7t{ exp-3 loop of nilpotence

class 3 and dimension (Hence, (S,*) c ^ since [S | = 312 .)

LEMMA. 6.2.6: (S,*) is commutative.

Proof: (yz)^ *= y£ + z ^ = z^ + y£ = (zy)^ for all e 1^ .

It was proved in the proof of Lemma k.2,6 that 7%^ z(t,i,j) =

^ #y(t,i,d) and so we know that ( y z ) ^ j = (zy)t j j for l

Next, we see that » ^ z(t,i,d;k, i) = cy (k,i) »

° y - Yksl> - (-c2,y(t» ( - (ziyjJ,k - zjVl» '

^ ^ y (t,i,j,k) j for all q e + ssi*q^V,2^t,J,k^ “

C2±Zq + y±yq)^ (t,j,k) ; and for all q e Ci,J,KJ ,

" f*,y(t)f*,y(q)^ , y (1';1'll) ' Henoe’

for all m € i^, we have < ^ 2(t,i,j,k) = o” (t,i,j,k) which

implies that (yz)' = (zy)' for all m £ I, . m m 4 Therefore, yz = zy . □

LEMMA 6.2.7: (S,*) is a quasigroup*

Proof: By Lemma 6.2.6 it suffices to show that given

z,x f S there exists a unique y £ S such that zy = x . If

such a y exists it must satisfy y = x - z for £ f I. , At At Xt H1

- xM , j - ^ . y ^ 1^ ) fOT (where the elements z and y = x - z determine W (t,i,j)), *■ A> At Xi ** y y and ~ xm ~ Zm " ^z, ^^>3,2,1) ^or m ^ ^4 Cohere the elements z., z, , .. y* = x* - z * * and y . . . = x . - z, . , -

77L (t,i,j) determine Q*" (^>3,2,1)) . And in fact, with y defined according to these requirements we see that zy = x . □

LEMMA 6.2.8: (S,*) is a loop with identity element

0 = [ 0,0,...,0] in which z"1 satisfies (z-^)), = -z.-z . for ' £ £ 226

i € 1^ 1 (z_1 )t,i,j = ~Zt,i,j for and

(z_1); - -z; for “ e \ •

Proof: For ail p and q with 1 < P < Q < *+, A _Cc3^P) = 0 z,0 which implies that Pi _(t,i,j) = 0 for l

With z-1 defined coordinatewise by the equations in the

statement of the lemma, we have A ^(q,p) = (z )(-z ) - z,z" p (Zp) (-Zq) “ 0 ald P and q with 1 < p < q < *+ . So

P\ _-|_(t,i,j) = 0 for l

LEMMA 6.2.9: (S,*) has exponent 3 *

Proof: Since (S,*) is commutative we need only verify that 2 — z z = 0 . For all p and q with 1 < p < q < 4 we have

6z,z(q' p) = (zq2p “ Zp V “ °* 30 “ ° f°r 1 - j < i < t < 1+ and pi „(t,i, j;k, X) = 0 for 1 < j < i < t < k and — Z|Z *“ — 227

1 < X < k < 4 . ^ ' * ' ' w ---- 2 2 M s o ' V ( M 'J'k) =

-zi for 1 € V + - -zt,i,d f°r

l

LEMMA 6.2.10? (S,*) is anfy loop.

Proof: We shall be concerned with the verification of the

^identity. So given x,z,y € S we let X = zy, p- = x(zy) = x\,

L = x[x(zy)] - xp,, a = xz, r = xy, and R - (xz){xy) = err • and we show that L = R .

We have X, = z^ + y< , ^ - X/ + z^ + y^, L£ . -X, + z£ + ,

°i = X 1 + V ' * t + y i ’ Snd R £ = -X £ + Z £ + y l ■ He”Ce L„ = E. for all I e I. . Next we have X, . . = z . + f I 4 t, i,j t,i,j yt,i,j + ’ “t.l.j * + Zt,i,J + yt,i,d + t,M) + , \ i)3 = + ytjl>j +

= x ^ . , + zt>i>j +

+ yt)1>;| + ^ y(t,l,J), and

+ zt,i,j + yt,i,j + + \ y4.i»j) +

T(b,i,j) . We have seen in the proof of Lemma 4.2.10 that

7% (t,i,j) ; therefore, L. . . = R. , . for l

Next we have for each m 6 I, ,

•i - * ; + + + ^ . w ) + < , x (4,3,2,d

*i - -xi + + + <,,(•‘.3.8.1) + <^,3,2,1) +

<,„(<►,3,8,1)

°m = xi + Zm * < , * C ‘'3.a.1>

Tn - + +

Bi ’ -xm + 2i + K + < ^ ( “.3,2,1) + <,,(*,3,2,1) -

< j T(1,3,8,1).

Hence, if we demonstrate that for all m £ 1^,

< , y d,3,2,D + < , x0».3,2,l) + <,„(>*,3,2 ,1 ) -

< , ZC*»3,2,1) + < , , ( * , 3,2,1) + < jT(1,3,2,1) then we shall have for a U m £ and the proof will be complete.

Because of the application this material will have in [1>1] we have defined Qr (t,i,j,k) for arbitrary values of t, i, j, and k rahter than just (h,3 ,2 ,l) ; for the same reason, we shall demonstrate that for each m € 1^,

< , y(t,1,3,1) + <^(*,1,3,1) + <,,,(*,1,3,1) -

<^(*,1,3,1) + < , , ( * , 1,3,1) + < (*,1,3,1) 229

I have verified by hand calculations that these four equalities

do in fact hold. Although these computations are perfectly

straightforward, they are extremely lengthy and can simply not

be included here in full detail, (of course, they can be

verified easily by a computer program which does polynomial

arithmetic over GF{3) •) I shall give a fairly detailed outline

of the computations which provides sufficient evidence that the

results are correct.

For elements z,y € S and positive integers p, q, s, v, w

we define the following four functions:

^,y(p,q,s'v’w) = V, W » ^,y(p'q’s’v'w> - 2pV s W *z,y(p'q’s'v'w) ’ 'pV«V» •

We see in the following computations that each of the

functions y (t,i,j,k) for m 6 1^ is a linear combination

of functions of the form 77^ s;v,w) and y(p,q, s,v,w) .

First, Q ^ y(t,i,j,k) = ^ (t,i,j;t,k) - !7^ y(t,i,k;t, j) +

Second, -(z^ + y ^ ) ^ y(t,J,k) - f ^ y (t)f^y (i);!^y (i, j,k) =

jS^y (t,i,i,j,k) - fi^y(t,i,i,k,i) - < ^ y(j,t,i,i,k) +

y(k,t,i,i,j) ; therefore, we have y(t,i,j,k) «

^ y (t,i,j;i,k) - ^ y(t,i,kji,d) + ^ y(i,3,kit,i) + 230

jjk) “ *5) " ^^y( +

*£,y(k,t,±,i,J* '

Next, we let *z#y(t,i,d,k) = -flz>y(t,j,J,k,i) +

^ y

^z + v (”k*ijj,j) - j£^ (t,j,k,i,j) - z>y z;y z>y

J#k#t,j) + $z^y (t,i, j,j,k) - 6~^y (t,j,i,jjjk) -

^ y t ^ d j ^ d j k ) +

^y(iik,t,j,d) -^>y(d,k,t,i,j) - j£ (i,t,j,J,k) . We find that Q ^ y (t,i,j,k) = ^ y (t,i,j;j,k) - ^ (t,j,k;i,j) +

j#kjt,j) + jOz^y (t,i,j,k) and y (t,i,,j,k) *»

5^ y (t,i,k;j,k) - ! ^ y (t,j,k;i,k) + ^ y (i,j,k;t,k) - fi^y (t,i,k,j) .

We let ^ D (p,q,s;v,w) = ^ y (p,q, s;v,w) + x(p,q, s;v,w) +

\ ^ ( p#qiSJv ,w) - (p,q,s;v,w) - ^ (p,q,s;v,w) -

T Cp,0., s ;v, w) and for each m f_ 1^, we let jS^(p,q, s, v,w) =

& v (P,q,B,v,w) + (p,q,s,v,w) + jtf“ (p,q,s,v,w) - “Jj A A, |X

^ z(p,q#s,v,w) - ^ y(p,q,s,v,w) - ^ T(p,q,s,v,w) . Using the following five relations we finally obtain that for all m £ 1^ and for all t, i, j, and k ,

<£)y(M>i,K) + -

fl£jB(t,i,J,k) - « ^ y (t,i,;),k) -

O O ( p , q ,H s , v , w ). l z p y q y s y v y v] + [ ( x p) ( z q yq) + ( z s + y s ) ( z v + w) y l vK*w+y + [ “ [ (^ + z )(x +y )(x +y )(x +y z )(x )] + + ; +y )(x +y )(x“ y [ (^ - [ x z z z z ] - [ x y y y v l + lV» - Vv -Vvl( + V M q V r ll(V v - v V V + [ = [v » - V v 1 I ( ,, y . - - . y ,, ( I 1 v V - » [v = x + -[x x lxz ] x y - - ]x z - [x l[x z -z -[x x x - [x+ + ^xp,q>s”zp»qq>s zsyq " ^zp ^ " yp ~ L p qs v L w p Lp" 7 q ' 7 s * 7 ^ wJ (x ,> pqs p s s s q p p p,q,s p,q>s ^^ pqsL wv p,q,s Jp,q,sJl- w vJ p,q,sJL w v P^q^s p P q v **w/J * w v s q q P P q q s vv w w w v v v s s s q q q p

,rv w w L v + + (x [x ) zy - (x x x p)VB-Vq,][(xv q + V zv)(xw - + + B y„)-(xw av,)(xv )(V yp - fxp + )Jyv )(x j w ’' 'v Jv w' v ’’v' w 1 + (z(x ) y -(zx x z y ( +xy xz xy )] y -x z )(x + -x z -y x y -z (x S P qs qJs qs p P P ,, p,q,s p,q,s +z + y = [ s p ) q > s - > P i , ) S ][v, - V z ( -z ( - ) - y - ) (x z - + -x x }(x z z 1 ( -Yq)(xp-V Y s- [(V + (V.'VqX-'j'V )(x -z + z +y - y - (z -y ) -y (z - y -z (z y y -

+z + )] +y s q s q q v ="(v " = +y q' p ,1 )(x ,, Jp,q,s p,q,s (p-Z V+ )(xp Zp - q +V V s^q' qs +z +y t T IT w *r

- x1 y x y )] v T ) j v - 231 15

4(® - t Z p V W w 1 + ((xp H x q)(z3 + ya)(zv + yv )(zw + yw )]

+ [ (xp ) (xq) (xs + zs + ya ) (xv + zv + yv ) (xw + zw + yv ) ]

-[xxzzz]-[xxy y v ] L p q s v w L p qJsJirw

- [(xp + Zp )(xq + Zq)(Xs + yB)(xv + yv )(xw + yw)] j

P»«.s,v,w) = fxpxqxsyvyw] + [(xp )(xq)(xs)(zv + yT)(zw + yw )]

+ [ ( x )(x ) ( x )(x +z +y )(x + z +y )] - [x x x z z ] L v p'v q'' s'' v v ■7v/v w w *'w/J L p q s v v

- [xp x q x sJry v w ] - [ L ( xx p+z p'' )(x q +z q')(x s +z s/v )(x v +y *'v/v)(x +w y Jw)1 /J * : and

K )(Xv)(ZW +yw )]

+ [Cxp )(xq )(x3)(xT )(xw + zw + yw )] - [xpxqxsxv2w ]

-[xxxxvp q s \rwJ ] - [ 1(x ' p+z p'v )(x q + z qyv )(x s +z s/v ){x v + z V )(x/v W +y *rW/ ) ].

DEFINITION 6.2.11; Ag = {z f= S : z » 0 for all z € I^J

Yl = 10,0,0,0,0,0,0,0,0,0,0,1]

y2 = [0,0,0,0, 0,0, 0,0, 0,0,1,0]

= [0,0,0,0,0,0,0,0,0,1,0,0]

" [ 0,0,0, 0, 0,0,0,0,1, 0, 0,0]

LEMMA 6.2,12; r ®= (Y ^ Y y Y2jYj_3 iB a basis of (S,*) and with r «= ( Yy Y2> Y^ , w have 233

A(4,3,2) = [0,0, 0,0, 0,0, 0,1, 0,0, 0,0]

A(^,3,l) = [0,0, 0,0, 0,0,1,0, 0,0, 0,0]

A(4,2,l) = [0,0,0,0,0,1,0,0,0,0,0,0]

A(3,2,l) = [0,0,0,0,1,0,0,0,0,0,0,01

and

B(lf,3,l;2,l) = [0,0,0,1,0,0,0,0,0,0,0,01

B(i*,3,2;2,l) = [0,0,1,0,0,0,0,0,0,0,0,01

B(^,3,2;3,l) = [0,1,0,0,0,0,0,0,0,0,0,01

B(^,3,2;^,l) = [1,0,0,0,0,0,0,0,0,0,0,01 .

Proof: We first note that for z,y € A , 277 (t,i,j) = 0 fa z,y for l

{[1,0,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0,0], ... ,

[0,0,0,0,0,0,0,1,0,0,0,011 .

Now,

~ [ o, 0, 0, 0, 0, 0, 0, 0, 0,1,1, o]

Y)+Y 3 = [0,0,0,0,0,0,0,0,1,1,0,01

Yj^(y3Y2 ) *= [0,0,0,0,0,0,0,0,1,1,1,01

(y^y3)y2 = [0,0,0,0,0,0,0,1,1,1,1,01 and

( y ^ Y ^ H O , 0,0,0,0,0,0,1,0,0,0,0] = [0,0,0,0,0,0,0,1,1,1,1,01 .

Hence, A(I+,3,2) = [0,0,0,0,0,0,0,1, 0,0,0,0] and we obtain the 23'+

analagous results for A(^,3,l), A(lj-,2,1), and A(3,2,l)

in exactly the same manner.

Next, we let Y = A(!+,3,2)y^ ;

Y^Y-l = [0,0,0,0,0,0,0,0,1,0,0,1] ,

Y = [0,0, 0,0, 0,0,0,1,1,0, 0,0] ,

and

a (1+j3,2)(yuy1) = [0,0,0,0,0,0,0,1,1,0,0,1] .

Since by (i,j) = 0 for 1 < 3 < i < 3 implies that * $ Y^

% v (t,i,,j) = 0 for l

7L (^,3,1,2) = 0 , and 7^ (p,q,s;v,w) = 0 unless v = U j *1 * Y1 and w = 1, we see that (**,3,2,1 ) = <£ (^,3,2,1) = 0 , *'Y1 x^Yi

<4 ,y 1 (1*’3’2' 1) ■ (i3,a,i ‘ = 0 • 0 = and

^ 3,2,1) = ( \ t3>2 - (v1)u^3^2) =1-0=1. Hence,

Yy-l = [1,0,0,0, 0,0,0,1,1,0, 0,1] and

( A f M ^ K Y ^ K l , 0,0,0,0,0,0,0,0,0,0,0] = Yyx = (A(U,3,2)^)Y;l5

that is, (A(4,3,2),yv Y1) = B(lf,3,2;4,l) =

[1,0,0,0,0,0,0,0,0,0,0,0] . The analagous results for B(U,3,2;3,1),

B(4,3,2;2,l), and B(^,3,l;2,l) are obtained in exactly the

same manner.

So, Ag £ < D •

Next we see that for all z f Ag and for ail e^, e^, e2, 6 6 © 6 and ex with 0 < e± < 2 for i € 1^ , ^ Y ^ C y ^ C Y ^ Y j 1))) - 235 z[0, 0, 0,0,0, 0,0,0, e^, e^e^] . This is analagous to Lemma

4.2.14 and follows from the facts that % (t,i,j) = e2 el Y2 'Y1

W ~ ^ = ° for 1 < J < i < t < 4 and 3 2 i _ _ y/,y2 y-l

L et e e e = A e e e e = 0 for Vif\Y 33(Y22Y11) Z,Y^(Y33(Y22Y11 ))

1 < j < i < 3 • Hence, S = (r) and S must be a basis since any three element subset of r generates a subloop of S having cardinality at most 3 * C

LEMMA 6.2.13: (S,*) has nilpotence class 3.

proof: By Lemma 6.2.12, (S, *) has nilpotence class at least 3j and since it has dimension 4, we know by Theorem 2.2.4 that its nilpotence class is at most 3* □ CHAPTER VII

MISCELLANEOUS RESULTS

§ 1. Designs Derived from HTS’s

We begin by considering those HTS’ s, S , whose coordinatizing

loops have center as small as possible; that is, |Z(Ge (S))| = 3 .

We have seen many examples of such loops: the Bruck loops Bg^

for i > 2 as shown in Proposition 3*1*1^ and the loops R^

for d > 5 as constructed in Theorem Jf.3.10 . We recall the definitions of central class and central line given in Definition

1 .6.1 .

PROPOS IT ION 7.1.1 (Young): If S = (X, fi) is an HTS and e € B 6 fi then B' € fi is ’’parallel" to B iff for some x 6 X \ B, B ’ <= Bx where Bx is a coset of B in Ge (S) .

Proof: We write B = (e,y,y **■] and consider B' = Bx = tx,yx,y~^x) for x 6 X\B. Bx is a triple of S: x o yx = x"^(y-^x-^") ~ y_1x . Bx is disjoint from B: x $ B does not permit yx = e, yx=y, yx=y-1 ,y -1 x = e, -1y x=y, or y~^x s y-*^ . B and Bx are coplanar: the plane of =

(X,fi,e) generated by B U (x) contains the points e, y, y-^", x, e a x s= x -1 ,yoX = y-1 x-1 , y o x-1 = y-1 x, y-1 <>x = yx-1 , and y"1 » x-1 = yx . Hence, B* = Bx is "parallel" to B .

236 Conversely, assume that B and B' are disjoint and

coplanar. Since the plane generated by BUB’ is isomorphic to

AG(2,3) , there exists a point x of this plane satisfying

x / B U B' and furthermore we must have B 1 = {e o x,y o x,y-^ c. xj , -1 -1 -1 -1, _ -1 lx ,y x ,yx J = Bx □

It follows from the above proposition that a central class

consists of central lines and that every central line is in a

unique central class.

PROPOSITION 7.1.2: If S = (X,B) is an HTS with |x| = 3m

and if |Z(G (S))| = 3 then S has a unique central class.

Proof: We have seen in the proof of the converse part of

Theorem 1-5.20 that for any e £ X, B = {e,y,e D y) is a

central line of S iff y € Z(G (S)) . So, the center of each

of the 3m isomorphic loops corresponding to S is the unique

central line of S containing the identity element of that loop.

Moreover, if aIff : G € (S) -* G W(S) is the isomorphism given in the -1 x *-> wx proof of Theorem 1.5.16 then the associator in Ge(S), (x,y,z) = e for all x,y 6 X iff the associator in G (S), (xCT,ya,zCT) = w for all x,y e X Iff the associator in Gw (S), (x,y,zCT) = w for all x,y € X . Hence Z(G (S))CT = Z(G (S)) . Since C W ol: e y, y <-*■ e , e 0 y y(e o y)-1 = e o y and a : j e o y e*+e»y, y ►+ (e « y)y-1 = y , e ° y ►+ e , we see that the central 238

line {e,y,e o y} is the center of the loops G (S) , G (S) , e y and G (S) . Thus, the number of central lines of S is e o yv ' 3m/3 = 3 ^ ” ^ » By the remark preceding the statement of this

proposition, there is a unique central class of 3 . This

central class is p = {Zx: x € X) where Z = Z(G (S)) . □

We recall the definition of a box given in Definition 1.6.1

and we also make the following definition.

DEFINITION 7.1.3: If S = (X, fi) is an HTS with unique

central class p £ fi then we define = (A = (X',fi') 6 £7(S):

B £ fi1 for some B € p) .

Keeping in mind that <7(S) is the set of all boxes of S,

we see that (7p(5) is the set of boxes which contain some central

line of S .

By Proposition 7 *1 .2 , the loop R^_ constructed in Theorem

U.3.10 corresponds to an HTS, S(R^_) , with a unique central class.

We let h be a generator of Z(R^) and e be the identity element of ; the ordered basis r = (y^f Y2j Yj_) of

Bqj , appearing in the construction of , satisfies

(V^jY^/Y2 ) = e • By the proofs of Propositions 7 .1.1 and 7.1.2 , any central line of S(R^) has the form B = Z(f^_)x where 0 0 g 0 0 x = he (Y^fYi/^Y^fYg2^ 1 )))) • That is, with y ** Y^Cy^ Ny^C^Y -j1)))) we have B = (h® y,he + 1y,he +2yJ j and of course, exactly two of the exponents e', e' + 1, and

e 1 + 2 are non-zero.

Since (Y^Yi^Vg) = e, we see that <( Y^/Y2) > =

o e e^ e^ th°(y ^(Y|^ Y2 )): 0 < e^t% * e2 — is a subgroup of order 27

which corresponds to a box containing no central line of S(R^) ,

So, in general <7.(3) / <7(S) although we have seen in Theorem

1 .6.2 that = tf(W8l) •

The following theorem is the generalization of Theorem 1,6.2 ,

THEOREM 7.1*J+: If S = (X, (3) is an HTS such that |X| = 3m

and |z(Ge (S))| = 3 then (S) | = 3m " 3(3m " 1 - l)(3m *2 - l)/l6

and given distinct 6 X , if (x^,x2,x1 ^ then x^

and x^ occur together in exactly (3m_1 - l)(3m _ ^ - 1)/16

elements of C7^(S) and if (x^x^x^ o x^) $ p then x.^ and

Xp occur together in exactly (3m **2 - l)/2 elements of <7JS) .

Proof: By Propositions 7.1.1 and 7.1.2, S has a unique

central class p given by p = (Zx: x € X} where Z = {e,h,h_1J

is the center of G (S) ; so £7_{S) is well defined. Note that e j? |pj = |Ge(S) | / |z| = 3m/3 = 3m ^ • We consider the linear

incidence structure S^ = (X,g) and recall that a plane of

is a subset of X which is a flat of dimension 2.

Let P = {TT t X: TT is a plane of and B c Tt for

some central line B C p) • For each TT £ P, TT = U U where B^,Bg,B^ € p; so we define an operation * on p by 21+0

by * Bg = B^ where is the third line in the parallel

class of and Bg in the affine plane determined by

and B2 . Note that Zx^ * ZXg *= Z(x1 o x^) since in the affine

plane determined by Zx^ and ZXg we have hx^ o hXg =

(h"1x^1)(h“1x21) - hfx^1^ 1) = hfx^oXg) and h ' ^ o h " 1^ =

(hx^Jfhx^1) = h“1(x"1x^1) = h-1(x1 oX2) .

Letting y = CCBpBg^B^} : B^,B2,B^ 6 p and for some

TT € P, TT=B^UB2 U B^J , we claim that S* = (^y) is an

STS in which every plane of the linear incidence structure

S^ = (pj^O is isomorphic to AG(2,3) . By definition, S* is

a 2 - (v *,k X *) design where v * = | p| = 3m " 1, k * = 3 S S S S S and X = 1 (note also that for this design r # = S S 1 * (3m ~ 1 - 1) _ 3m ~ 1 - 1 and b _ 1 -3m ~ 1 • (311 ~ 1 - 1) _ 2 2 d b * “ 3 *2

_m *2/“ i _ -i ^ * *----^ ~ 2 ------) 5 that is, S is an STS. The operation *

* defined above is precisely the quasigroup operation of S .

Now since S is an HTS, we Know by Proposition I.5 .I5 that

Zx1 * (Zxg *Zx^) = Zx1 *Z(Xg o x^) = Z ((x^ 0 (xg o x^)) =

Z ((x1 « Xg) - (xx o X3)) = 2 ( ^ 0 ^ ) * Z (x1 o x3 ) =

(Zx^ *ZXg) * (Zx^ *Zx3) • Hence, * is distributive and by

Proposition I.5.15 we Know that every plane of is isomorphic to AG(2,3) ; so our claim is proved. . *. Note also that the loop G^(5 ) with operation written as • is isomorphic to Ge(S)/Z: Zx^ * Zx^ «= (Z *Zx^) * (Z *ZXg) = 21*1

-1 „„ -1 -1 -1, z(e o *Z(e » x^) = Zx^ *Zx^ = Z(x^ » x^ ) « Z x ^ '

-tt* -tt- Now a plane TT of has as its points nine central -£ lines of S ; letting $(TT ) = (J B , we see that the sub- B eTT*

-Jf. system of S with point set $(tT ) is a box of S and is in

fact a member of CfJs) . Furthermore, for each box A £ <3 A S) it t -tt # -tt* there is a unique plane TT of such that $(TT ) is the -tt -tt* -tt* point set of A . Hence, |c?^CS)| = 11 TT : TT is a plane of S^J J .

Let T = [ ({B, Y),TT*): B e p, Y € % B i Y, and TT* is a plane of S with (tB) U Y) = TT*] . Given one of the b = € 5 m - 2 , m - 1 , *------~— triples Y £ y t there are v x. - 3 =3(3 ” -1) S elements B € p with B $ Y . B and Y determine the unique plane TT = (tB) U Y) j hence, we have on one hand that |t | =

m - 1, m - 1 _ . u „m -2 _ , ----“ ----- v------*- . Given one of the \<3 (S) | planes d t -tt- -tt- -tt- -tt- TT of S^, there are twelve lines, Y, of with Y c IT since TT is isomorphic to AG(2,3) . And there are 9- 3= 6 * j points, B , of TT with B ff Y . Hence, on the other hand

(T| = I^JS)) • 12 • 6 and we conclude that |c7 (S)| = it it

3m -lf3m - 1 - 1) f 3m ~ 2 - 1) 3m - 3 f,ra-l _ ^ 2 • 12 • 6 “ lS

Next, for each Y £ y let t'(Y) = |{TT*: TT* is a plane of

S* and Y = TT*] | and let T ' = ((B, TT*); B € p, B £ Y and

TT* = } . We have |T' | = (3m "1 - 3) • 1 and |T' | = t'(Y) *6 which implies that t'(Y) » (3m ~^ - l)/2 • Note that t ’(Y) ifi independent of Y. Next, for B € P let t"(B) *= |{TT*: TT* is a plane of

and B € TT*} | and let T" = | C CB1 j TT^): TT* is a plane of

S*, B 1 / B, and B,B' € TT*} | . We have |T” | « (3m-1 - l)t'(Y) ■* where Y is the line of S determined by B and B' : and t on the other hand |T" | *= t"(B) *8 . Hence, t"(B) =

(3m ~ x - D t g ” ' 2 - D / a (3111-1 - 1^ m ' 2 - J-) .

Now, if and Xg are distinct elements of X and if

(x^,Xg,x^ o x^) £ p then any member of £7 (S) which contains x^ and Xg contains the central line {x^,Xg,x^ o Xg) ; and by the

/^m-1 _ _ m - 2 ^ . above computations there are — .. ■ ■ ------*• such boxes.

If {x^,Xg,x^ 0 Xg} p then the central lines Zx^ and ZXg determine the plane of with point set Zx^ U Zx^ U (Zx^ *Zx^) ,

And since Zx^ *ZXg = Z(x^ o Xg) , the line (x^,Xg,x^ o Xg} is a line of this plane. So, with Y = (Zx^,Zxg,Zx^ o ZXg} we know from our above computations that x^ and Xg are contained together in t'(Y) = (3m_2 - l)/2 elements of £7p(S) . □

Note that in other language (see [Tj-9]) Theorem 7.1.U states that (X,£7^(S)) is a (v,b,r,k, *g)-partially balanced incomplete

m - 3, m -1 m -2 * block design with v = 3 , b = ^ I'g'------~— ^ t 243

We now turn to the designs which can be derived by viewing

an HTS as a rank 4 PMD and applying the general constructions

of Young and Edmonds found in Theorem 1,4.20 . Given an HTS

S = (X, B) (and its corresponding linear incidence structure

S€ = (X,8,€)), with |X| = 3m we let Jq = (0) , ^ = P^X) ,

J>2 = P2 (x), J?3 = P3 (X)\B, = {(x1 ,x2 ,x3 ,x1;} e r^(x): I* Is not a plane of S^} , and JQ = U ; k = 0

then Mg = (X,Jig) is a rank 4 PMD in which the O-flat is 0,

the 1-flats are the points of X, the 2-flats are the triples

of d 3 the 3-flats are the nine point affine planes of S^,

and the 4-flat is X .

We recall Definition 1,4.19 and we have the following result.

PROPOSITION 7-1.1;: If S = (X,fl) is an HTS with

jxj = 3m and Mg is its associated rank 4 PMD then c(Mg) = 3 ,

- 8, - 1, *3 - S O * - 1 - l), * - 33O m - 1 - l M 3 m -2 - l ) )

X3 . lf x, . and X;- =

Proof: By definition of Mg we have aQ = 0, a = 1, a2 = 3, a3 = ab ~ c^ ) - 3, and p(Mg) = 2 . And so by Definition

a, - a 1.4.19, we have ^ = 1, ^ ^ = 3m - 3 = 3(3m - X - 1) ,

_ (°4 - °a)(^| - a3> (3m - 3 ) O m - 32 ) 33(3m - 1 - l ) O m -2 -l) (4 - 2)1 2 2 * 2kk

ch + (-l)1(?)o;, - r ^ 1 l q / ^ \ t \ _ *— M21 ^ _ 2 * ^ “ W "2) — (3 " 3) ” ^ ;

ah + (C-l)2C^)c»1 + (-l)1^ ) ^ ) \ = k - 2 ^ " Mtf

. SO” -1 - l ) [ 3 m + j -32] - £ » m ~1 -l)(3m-2 -l)

= £i£ii^iW-i + 1.3). (3-i.3)] ^u l ; 1-1),

and

% + (C-D^i^ + (-D2^ ^ + (-D1(^K) ^ - M ^ [ ------5 - ^ 2 ------

= 33(3m~X - lK3m "2 - 1) [3m + f-tf + 18 - 3 6 ) ]

_ 32(3m“X - l)(3m“a - l)(3m - 2 2 )

PROPOSITION 7.1.6; If S = (X, 13) is an HTS with |X | = 3m ,

Mg is its associated rank 4 PMD, and t = t is the t-function of Mg then

t(0,l,2) = 3

t(0,l,3) = 9

t (0,1,it) = 3m

t(0,2,3) = 12

m - l ( m , t(0,2» = ^ W --- ±1 2

m - 2, _m , w ,,m -1 £1*5

t(l,2,3) = 1*

t(i,2,.*).

-3m " ^ T t(2,3A) = 3-- 5-=^-

Proof: By definition of the t-function, t(0,l,i) = ;

hence, t(0,l,2) = 3, t(0,l,3) =9, and t(0,l,4) *= 3“ . By

Theorem 1.U.13,

t(2 3 1*) = tjO.l.1*) - tjo.l.g^ _ 3* -3 3111 " 1 - 1 1 ,:i’ ; t(o,l,3) - t(o,l,25 9-3 2

t ( o 2 ,3 ) = 3 ) = 2— lit = ip ' ' >3} t ( o , i , 2 ) 3

t ( 0 A u) = . a ? m - 1)/2 _ i)

t(lf3fh) m ^ ^ .) t ( ^ 3 j U ) = l ) /2 - l ) /2

_ (3m - D i f ' 1 - 1) 16

- l l %

3m ~ 2(3m - ^ □

Note that the conclusion of the above proposition also holds when = (X,B,6) is isomorphic to the point-line incidence 246

structure of AG(m, 3) and that these results can be applied * to the STS S in the proof of Theorem 7.1.4 in lieu of the method of "counting in two ways" to yield a shorter proof. (Of course, Theorem 1.4.13 itself is proved by "counting in two ways" and I wanted the proof of Theorem 7.1.4 to be self- contained. )

The following theorem is an immediate corollary of

Proposition 7*1*9 Qnd Proposition 7*1*6 together with parts (i),

(ii), (iii), and (iv) respectively of Theorem 1.4.20. Recall that 3. denotes the set of i-flats of a PMD and C. denotes i 1 the set of circuits of cardinality i of a PMD.

THEOREM 7*1*7: If S = (X, b ) is an HTS with |X| = 3 ™ and Mg is its associated rank 4 PMD having t-function t = t then “s

(i) If x € X , 3^ = tB 6 fi: x e B}, 3^ = (TT € 33: x € TT] , and y* = t EB € l£: B = TT] : TT € 3^) then DX(S) = (3^7*) is a ((3m - l)/2, 4, 1J-BIBD .

(ii) (X,33 ) is a (3m, 9, (3m - 1 - l)/2)-BIBD.

(iii) (X,J^) is a (3m, 3, 3C3m “1 - 1))-BIBD and

(X,^) is a (3m, 4, 33(3m_ 1 - l)(3m " 2 - l)/2)-BIBD.

(iv) (X,C^) is a (3m, 4, 32(3m “ 1 - l)/2)-BIBD and

(X,<^) is a (3™, 9, 32(3m - 22)(3m " 1 - l)(3m “ 2 - l)/2)-BIBD . 2kJ

Note that the circuits of are sets of four points of

X , no three of which are a triple of ft but which lie in the

same 3-flat of NL ; and the circuits of C*. are sets of five

points of X no four of which lie in the same 3-flat of M- .

Also note that is the complementary design of S ; that

is, = P (X)\B *

Because of the large values of X in the designs constructed

in parts (ii), (iii), and (iv) of Theorem 7 .1.7, the designs

D^fS) are probably the most interesting of the designs derived

from HTS's. ° ^ en called the local design at x and we see that a local design of an HTS of order 3m has the parameters of the point-line incidence structure of FG(ra-l,3) .

We shall use a result of Doyen and Hubaut (see [19]) to show that these local designs are not isomorphic to the point-line incidence structure of PG(m-l,3) •

DEFINITION 7.1*8: If T = (P,£,€) is a finite linear incidence structure we call the partially ordered set (3,c ) the linear space of r where 3 is the set of non-empty flats of r * For p 6 P the local space at p is denoted by and defined as the partially ordered set (3P,s) where

3? = { F € 3:p€F and dirn(F) > 1] . (That is, aP consists of the lines on p, the planes on p, etc.) r is said to be locally projective iff for some fixed d and q , D^ is isomorphic to PG(d,q) for every p € P • T is said to be k-regular iff |l| = k for all L 6 £ * r is said to be 248

d-dimensional (note that this term is given a special meaning

here which is much more than just dlm(p) = d) iff d > 2

and for all j with 0 < j < d, there exists 3. c 3 such J that 3O = p, 3, ^= £, 3, ti = (P), and given P f 5. J and p € P\F, there exists a unique element F* € 3. , satisfying j + 1 p 6 F 1 and F £ F ’ . T is a Lobachevsky space of type m

iff m > 1 , r is d-dimensional, and for all p g p and for

all L € £ such that p / Lj |(L* £ <(p} U L): p 6 L 1 and

L ' 0 L = 0} | = m .

THEOREM 7*1.9 (Doyen and Hubaut): If T = (?,£,€) is a k-regular, locally projective linear incidence structure with dim(p) > 3 then r is an affine space, r is a projective 2 space, or f is a Lobachevsky space of type k - k + 1 or type k + 1 .

THEOREM 7.1.10: If S = (X,fl) is an HTS with |X | = 3m then for x 6 X the local design D (S) is not isomorphic to the point-line incidence structure of PG(m-l,3) .

Proof: Note first that Aut(S) is transitive on X : given e,w € X, the isomorphism i|c Gg(S) -> Gw (S) of Theorem

x y* wx_1

I.5 .I6 is an automorphism of S with e^ = w since x^ o y^ s= wx-1 o wy'1 = (w-1x)(w”1y) = w_1[ w-1(xy)] = w(xy) = (x-1y-1)^ “

(x0y)^ . Hence, for all x,y € X, D (S) is isomorphic to D (S) . x y If Dx(s) were isomorphic to the point-line incidence

structure of PG(m-l,3) then every local space of the linear incidence structure * (X, (13, €) would be isomorphic to

PG(m-l,3) and would be 3-regular, locally projective, and would satisfy dim(X) > 3 . So, by Theorem 7.1.9, would have to be a projective space, an affine space, or a

Lobachevsky space of type 7 or of type 28. Since none of these possibilities is the case, D (S) is not isomorphic to the point-line incidence structure of PG(m-l,3) , G

We shall now give the method for obtaining the local design

V*81 ) which has the parameters of the point-line incidence structure of PG(3,3) ■ (This method will work for any HTS whose corresponding c-5?f exp-3 loop can be constructed by the method of Theorem k.2.5 ,) We first recall from the paragraph immediately following Corollary 4-.2.15 that Ge(Mqj) = G8l can be represented as GF(3)^ where for z = (z',z^,z2,z^) and y = (y’,y3,y2 >yi) the operation * of GQl is defined by z *y ■> (z1 + y' + 7^, (3,2,1 ) , z^ + , z2 +y2 , z ^ y ^ .

(Recall that 2 ^ y (3,2,l) = {z^ - 2,1) where A ^ y (2,l) z2yl " zly2 We write e = for the identity element of G8l .

We have 3® = [ ( (0,0,0,0), (z ' ,z3,z2,z1 ), (-z ’, -z^ -z2, -z.^} :

(z\ Z y Z ^ tz^) € GF(3)^ \ ((0,0,0,0)]} . Given two points of 2^0

the other two points of De^*8l^ 0n the line of Ce^*8l^ through z and y are the other two lines of through e in the plane of *81 determined by z and y : namely,

(e,z-1 » y'^e o (z"1 o y-1)) = = ((0,0,0,0),

(z* + y’ + ^ ^ y(3,2,l),z3 +y3,z2 + y2,z1 + y1),(-z* -y' ^ ^(3,2,1), z ,y "Z3 -y3,-Z2 “ y2,-zl "yl ^ and (z’1 ^ ) ) =

(e,zy-1,z_1y) = ((0,0,0,0), (z • -y* + %{ (3,2,1),z - y ,zg -y z,y Z1 ~ yx)j (-z 1 + y' + #r (3,2,1 ),-z3 + y3,-z2 + y2 ,-z1 + y1)J . z , y In the following list (z 1 ,z3,Zg,z^): i means that we shall denote by the numeral i the line of through e == (0,0,0,0) and (z',z ,2,,^^) j such a line will be a point of De(Hg^) •

(0,0,0,1 ): 1 (0,1 ,2 ,0 ): 11 (1 ,0,2,1 ): 21 (1,1,2 ,2 ): 31

(0,0,1,0 ): 2 (0,1,2,1 ): 12 (1 ,0,2,2 ): 22 (1,2 ,0,0 ): 32

(0,0,1,1 ): 3 (0,1,2 ,2 ): 13 (1,1,0,0 ): 23 (1,2,0,1 ): 33

(0,0,1,2 ); k (1,0,0,0 ): Ik (1 ,1,0,1 ): 21+ (1,2 ,0,2 ): 3^

(0,1,0,0 ): 5 (1,0,0,1 ): 15 (1,1,0,2 ): 25 (1,2,1,0 ): 35

(0,1, 0,1 ): 6 (1 ,0,0,2 ): 16 (1,1,1,0 ): 26 (1,2,1,1 ): 36

(0,1,0,2 ): 7 (1 ,0,1,0 ): 17 (1,1 ,1,1 ): 27 (1,2 ,1,2 ): 37 (0,1 ,1,0 ): 8 (1,0,1,1 ); 18 (1 ,1,1,2 ): 28 (1 ,2 ,2 ,0 ): 38

(0,1,1,1 ): 9 (1,0,1,2 ): 19 (1,1,2,0 ): 29 (1,2,2,1): 39

(0,1,1,2): 10 (1,0,2,0): 20 (1,1,2,1); 30 (1,2,2,2): 40

With this notation we find the ho • 39/k * 3 - 130 blocks of

^e(*8l) ^hose ^-tuples listed in Table 5 . We abbreviate

(i,j,k, £} by i j k I. 251

Table ; The Blocks of 5 : De < % >

1 2 3 4 2 5 8 11 3 6 29 3 9 4 7 27 3 5 5 1 7 2 6 3 5

1 5 6 7 2 6 3 0 4 o 3 7 2 6 3 7 4 8 2 4 3 6 5 18 2 7 3 6

1 8 27 3 9 2 7 28 3 6 3 8 3 0 3 3 4 9 2 9 3 3 5 1 9 28 3 7

1 9 2 8 3 8 2 9 2 4 3 7 3 10 2 4 3 5 4 11 25 4 o 5 20 2 9 3 8

1 10 2 6 4 0 2 10 3 1 3 3 3 11 28 3 4 4 1 3 26 3 4 5 21 3 0 3 9

1 11 3 1 3 7 2 12 2 7 3 4 3 12 2 5 3 8 4 l 4 1 9 21 5 22 3 1 4 o

1 12 2 9 3 6 2 1 3 25 3 9 3 1 4 18 22 4 1 5 1 7 22 6 8 1 3 1 9

1 1 3 3 0 3 5 2 1 4 1 7 20 3 1 5 1 9 20 4 16 18 20 6 9 12 1 7

1 1 4 1 5 16 2 1 5 18 21 3 16 1 7 21 4 2 3 2 8 3 0 6 10 11 18

1 17 18 1 9 2 16 1 9 22 3 2 3 27 3 1 4 3 2 3 7 3 9 6 l 4 2 4 3 4

1 20 21 22 2 2 3 2 6 29 3 32 3 6 4 0 5 1 4 2 3 3 2 6 15 2 5 3 2

1 23 2 4 2 5 2 3 2 3 9 3 8 4 5 10 12 5 1 5 2 4 3 3 6 16 2 3 3 3

1 32 3 3 3 4 3 5 9 1 3 4 6 3 1 3 8 5 16 2 5 3 4 6 20 2 7 3 7

6 21 2 6 3 6 8 15 28 4 0 10 1 7 2 5 3 6 12 10 26 3 3 16 3 8 3 9 4 o

6 22 28 3 5 8 1 7 29 3 2 10 1 9 3 0 3 2 12 1 9 2 3 3 9 1 7 2 4 27 3 0

7 8 12 22 8 1 8 2 5 3 7 10 21 2 3 3 7 12 20 2 4 4 0 1 7 3 4 37 4 0

7 9 11 21 8 20 2 3 3 5 10 22 2 9 3 4 12 21 2 8 3 2 18 2 4 2 8 2 9

7 10 1 3 20 8 21 3 1 3 4 11 12 1 3 1 5 1 3 1 4 3 1 3 6 18 3 4 35 3 9

7 1 4 25 3 3 9 1 4 2 7 4 0 11 1 4 29 3 5 1 3 16 2 9 37 1 9 2 4 2 6 3 1

7 15 2 3 3 4 9 1 5 26 3 9 11 16 3 0 3 6 1 3 1 7 2 8 3 3 1 9 3 4 3 6 3 8

7 16 2 4 3 2 9 18 3 1 3 2 11 1 7 2 3 3 8 1 3 18 2 3 4 o 2 0 2 5 28 3 1

7 1 7 3 1 3 9 9 1 9 2 5 3 5 11 1 9 2 7 3 3 1 3 21 2 4 3 8 20 3 3 3 6 3 9

7 18 3 0 3 8 9 20 30 3 4 11 20 26 3 2 1 3 22 2 7 3 2 21 2 5 2 7 2 9

7 1 9 2 9 4 o 9 22 2 3 3 6 11 22 2 4 3 9 1 5 2 9 3 0 3 1 21 3 3 35 4 o

8 9 10 16 10 1 4 2 8 3 9 12 1 4 3 0 3 7 1 5 35 3 6 3 7 22 2 5 2 6 3 0

8 1 4 26 38 10 1 5 27 3 8 12 16 3 1 3 5 16 26 2 7 28 22 3 3 3 7 3 8

We verify Theorem 7 .1.10 for Wg by noting that the triangle

{1 ,2 ,5 } of l^e f *^8 1 ) &enerates the entire design, rather than a thirteen point plane isomorphic to PG(2,3) . Also note the 2^2

correspondence between the boxes of *31 containing e and

the subsystems of D( i ^ W8l ^ which are isomorphic to PG(2,3) ;

for example, the triangle (1,2,1*0 generates such a plane

and corresponds to the box of % generated by e, (0,0,0,1),

(0,0,1,0), and (1,0,0,0) .

Theorem 7*1*7 can be applied to all the HTS’s which have

been constructed for the first time in this treatise and I

believe that these designs, particularly the local designs, are

worthy of future study.

We close this section by mentioning that the process of

"erecting" an HTS from a rank 3 PMD (recall that every (v,k,l)-

BIBD is a rank 3 PMD) to a rank ^ PMD - declaring the "new" rank

3 flats to be the affine planes generated by all the triangles -

can not be continued to obtain PMD’s of rank greater than four.

PROPOSITION 7*1*11: If S = (X,B) is an HTS then it is

impossible for every tetrahedron of S to generate a box of S .

Proof: If every tetrahedron of S generated a box then

by Theorem 1.5.20, for each e e X and x,y,z 6 X, x, y, and

z associate in G (S) . This implies that the loop Gg(s)

would be associativej but since S is an HTS, this is impossible

by Corollary 1.5*21 . □

PROPOSITION 7.1.12 : If S = (X,B) is an HTS then it is

impossible for every tetrahedron of S to generate a subsystem isomorphic to . 253

Proof: The conclusion follows immediately from part (iv)

of Theorem 1.2.6 which guarantees that every subsystem

S" = (X",fl") of S, generated by a tetrahedron, has a subsystem

S' = (X',tfJ') which is a box generated by some tetrahedron of S . □

§ 2, Automorphism Groups

We have mentioned in § 6 of Chapter I that if S = (X, ft)

is an HTS and = (X,ft,6) is its associated linear incidence

structure then F(sl £ Aut(S) where F(S-) =

It is likely that these Fischer groups will be the most interesting

groups associated with HTS's and a good topic for future

investigation will be the determination of the cardinality and

the structure of these groups. We have seen that by Hall's

construction given in the final paragraph of §2 of Chapter I,

the Fischer group of has cardinality 2*3' and

contains the Burnside group B(3,3) as a normal subgroup of

index 2 . We mention in the following proposition that minimal

generating sets of F(S^) correspond to minimal generating sets

of X . (Of course {e,x1,x£,...,xd) is a minimal generating

set of X iff (x^^Xg,...,xd) is a basis of Ge(S) ; so, every minimal generating set of X has the same cardinality by

Theorem 2.3.8 .) PROPOSITION 7-2.1: if S = (x, 6) is an HTS with associated

linear, incidence structure = (X, fi, e) then X = {(x^x^,,.. .,xd) >

iff the set {a ,a , ...,a ) generates F(S ) . Xi Xg xd £

Proof: It suffices to show that x o y = z iff a a a - a . x y x z a a a a a a First assume that a = a a a : z x y x = z iff (z x y) x = z z x y x * v 1 a a a XV X iff z = z since a is an involution. Since a fixes x y a only y we must have z = y , which is to say that x « z = y which is equivalent to x o y = z . Second, assume that x o y = z . We have a - (x)(y,z).. a = (y)(x,z),.., and a = x y 2 a aaa a aaa (z)(x,y)»-* so certainly xz=y = x xyx, yZ =x=yxyx, a aaa Z X V X and z - z = z J . Now, if p £ X\ tx,y, zj we consider the affine plane of generated by x, y, and p: we can write the lines of this plane as (x,y,z}, (p,q,s}, (t,v,w},

(x,p,t), {y,q,vj, {z,s,w}, {x,q,w), ty,s,t), [z,p,v), (x,s,v),

{y,p,w}, and Cz,q,tJ . We have

= (x)(y,z)(p,t)(q,w)(s,v)...

«y = (y)(x,z)(q,v)(s,t)(p,w)...

a = (z)(x,y)(s,w)(p,v)(q,t)... and

axay«x = Cz)(x,y)(s,w)(p,v)(q,t)... .

az aaa In particular, p = v = pxyx; hence, az = . □ We now make a very modest beginning of the study of the full

automorphism groups of HTS's; in particular, we shall determine

the elements of Aut(Fg d) where Fg d is the free c-D\ exp-3

loop of nilpotence class 2 and dimension d constructed in

Theorem **.3*1 . With S(Fg d) = (X^,fld) being the HTS

corresponding to Fg d , we know by Proposition I.5 .I9 that

Aut(Fg d ) is a subgroup of Aut(S(Fg d )) and in fact is the

stabilizer of e € Xd where e is the identity element of

Fg d . This will enable us to compute the cardinality of

Aut(S(Fg^d )) .

We begin with two lemmas concerning bases of Fg d •

LEMMA 7 *2.2; We let T = (x^,Xg,...,xd) be a d-element subset of F„ , : then r is a basis of F„ , iff for all 2,d 2,d 1 € id , x. i .

Proof: By Theorem 2 .3.0r , it suffices to show that T * =

{XiFg ^j^Fg dJ **’,XdF2 d^ is a basis of the elementary abelian

3-group F2,d/F2,d iff f0r a11 1 € Zd * x. ^ . 1 J First assume that for some i 6 I , xi ^ F2 d U *Xj: 1 < j < i - 1} ) . Since F^ d = Z(Fg^d ) we

^i — 1 ei — 2 eg can write x^ = ^xi - 2 xi ) - - * )) J®' where ej € {0,1,2) for 1 < j < i - 1 and g' € F^ d . Hence, x^g d

6 0 € G [xii_^1 (xii_g2 (...(x22xi;L)...))]F^ d which implies that V* is 2^6

not a basis of F0 ,/f * , . 2,d' 2,d Second, assume that r is not a basis of Fp h/f * . y U C , y U i Since d(F2jd/F'jd) = d (F ^d) = d and in feet = C* , * , we know that r is not an independent set in Fp ,/F* , . ( 1 £ - y Q . eo That is, we can write (x-jF^) (^ F ^ ) ***(xdF^ d ) = F2,d

where e. £ (0,1,2} for all j £ I and for some i € I, , J * d d •* ^ 0 . Let k e= max (i: e^ / 0} ; we see that

e e © s [xkk(xkk_^1(a..(x22x11 ).a.))]F^d » d . In particular,

6 6 6 6 xkk[xkk_"11(... (x^x^ ) ...)] = g' for some g' € F^ d . Since

etr _ ^ _ 1 _e2 ”ei F2 d has the inverse property, x^ = [xfc_1 (...(Xg )...)Jb '

and since ^ 0 , ^ £

LEMMA. 7 .2.3: If r = (xd, xd _ . .,x1 ) and

v = (yd,yd_ . . *>yd) a**e ordered bases of Fg d then the

mapping i|r: x^ -*■ y^ can be extended uniquely to an automorphism

t of F2 d ; that is, there exists a unique element

7 £ Aut(Fp ,) such that 'txd,xd _ j, ,. *,x^j , = i)r .

Proof: By Lemma 3*1*13* every element z £ Fp , has a d., a

0d °2 01 unique representation as z = g[ xd (...(Xg x^ )..,)] with

g £ Fg d and £ (0,1,2} for all i £ Id . If | is an automorphism of Fg d extending + then necessarily z^ =

7 ed e2 el gf[yd (...(y2 )•••)] • Furthermore, we have z. . . w i n g *= TT (x,,x ,x_) 1 f0 j hence, we see that necessarily l

J = TT ((x x x = l

z . TT (y+;y.,yJ ,1; , the last equality following from 1 < j < i < t < d " the definition of the associator. In particular, we must have |V (xfjx4>xJ = (y+,y^,y.) for l

): l

F2 d * f°r examPle> x^,x^,x^ and were elements of two bases of a loop and (x^,Xg,x^) <= e and £ e 1 then there could be no automorphism i|r of the loop satisfying tjr: ^ yj_ for aU- i € I3 *)

Recalling the proof of Theorem k.l.J, we see that i|r as defined above is an endomorphism of Fp , (and hence, an 2 , a automorphism of Fp , since it is a bijection): with ^ j a — 2 2 2* Z we have z+ « TT (yt,y±Jy.) * y,d[... (y/y,1)...] , 1 < j < i < t < d x 1 J d * 1

— W . W V w = Tt (y+^yi,y.) * y.d[ ...(yP2y11 )...] , and l

¥ T i i + wt i i z V = tt (y^y^y.) ' 'J ' 'J 1 1 < j < i < t < d J

Z + W Z y + W0 z + w (yd [«••(y2 y2 )•••])

= (Z W ) ^ . □

THEOREM 7 . 2 A ; If T = (xd*xd is a fixed

ordered basis of Fp , then the following two statements hold. 2,a

(i) If o e Aut(F2 d ) then for all i f

Xi ^ (F2,d U {xy 1 < J < i - U > •

(ii) If a: {xd,xd _^ ..,x } F2 d is a napping such that

for all i € Id , xi ^

then ct can be extended uniquely to o € Aut(Fg d ) .

Proof: If a 6 Aut(Fg d) then (x^x^ is a basis of Fp , and by Lemma 7*2.2, the conclusion of part (i) is 2,a immediate.

If a is a mapping which satisfies the hypotheses of part

(ii) then by Lemma 7.2.2, (xd*xd is 6111 ordered basis 259

of Fg d ; hence, by Lemma 7 .2 ,3 , a can be extended uniquely

to a 6 Aut(Fg d ) • □

COROLLARY 7 .2.5 : For all d>3, if (xd,xd _ .. .,x1)

is an ordered basis of Fg d then Aut(Fg d ) =

(ct: a: [xd,xd * fx^} -> Fg d is a mapping such that

Xi &

particular, |Aut(Fg ,) | = 3^^^ * TT (3d - 1) where > i =1

Y(d) = d(d ) + (d) .

Proof: The first statement is immediate from Theorem 7 .2 .14- .

Next, since for all i € ld , x^ (Fg d U (xj: 1 < 0 < i - 1J ) ,

we see that for all i r € {0,1,2,..,,d - 1} ,

d ) + i' J

d-1 d+(^) (n ) + 1 '

of this corollary, |Aut(Fp j ) | = TT (3 - 3 ) = ' i ’ =0

d - 1 (d) + i* d - 1 d(d) d - 1 , d rr 3 3 • TT (3 - 1) = 3 * TT 3 ' TT (3 - 1) = i ' = 0 i'=0 i’=0 i = 1

d(3> <2> d i Yfd> d i 3 *3 • TT (3 - 1 ) = 3 { ) * TT (3 - 1 ) . □ i =1 i = 1

COROLLARY 7.2.6: If S (F2 d ) = (Xd>Bd) iS the HTS corresponding to the c-ty exp-3 loop Fg d then |Aut(S(Fg d))I =

3A(d) t rr (31 - 1) where A(d) = (d + l)(d ) + (d) + d . i = 1 5 Proof: We shall use the following standard re stilt from

permutation group theory (see [3])- If (X,G) is a finite

permutation group then for x £ X, |&x( = |G | / |Gx | where

©x “ fx®: g € G] is the orbit of x and Gx is the stabilizer

of x (recall Definition I.5 .18).

Since the permutation group (Xd,Aut(S(Fg d))) is transitive

, . d + ( 3 5 |®el = ls (^2 d ^ = ^ ’ hence, by the above result,

, d+^> Proposition 1 .5 .19, and Corollary 7 .2 .5 , )Aut(S(Fg d ))j = 3

|Aut(F2 d)| = 3 ^ • 3Yfd) • TT (31 - 1) = 3A(d) • TT (31 - 1) ' i = 1 i = 1

since d + (d) + Y(d) = d + (d) + d(d) + (d) *= (d + l)(d) +

(d) + d. □

ri 1 o COROLLARY 7 *2 .7 : |Aut(j^)| = 2^ . 3^ . 13 .

Proof: We have seen that Gg^ is isomorphic to Fg ^ ; therefore, is isomorphic to S(Fg g) . So by Corollary

7 .2 .6 , (Autt^)] = 3k *1 4 3 + 3 . 2 *8 *26 = 2^ . 3 10 *13 • □

Recalling the well known fact (see [17]) that |Aut(AG(m,q))|

m m " 1 m i m+^ m i q • TT (q-q)=q * TT (q - 1) , we see in particular i - 0 i =1 that |Aut(AG(^,3))| = 3 10 • 2 • 8 • 2 6 • 80 = 2 9 • 3 10 * 5 .1 3 . T h a t the automorphism group of % 1 is somewhat smaller than the automorphism group of the eighty-one point geometry AG(U,3) 26l

reflects the fact that is somewhat less regular than

AG(^,3): some tetrahedra of generate boxes and some do

not whereas all tetrahedra of AG(4,3) generate boxes.

I have learned from a private communication from Jill

Yaqub that some knowledge of the structure of Aut(Wg^) can

be gleaned by examining the pointwise stabilizer of a box and

the action of a Sylow 13-subgroup on the set of twenty-seven

central lines of and on the set of thirty-nine boxes of

. It is my hope that these techniques can be combined with

some of the powerful theory of permutation groups as given in

Wielandt's book [ 63] to completely determine the structure of

Aut(Wg1 ) and the automorphism groups of other HTS’s as well.

§ 3 * Concluding Remarks

In the study of HTS's undertaken in this treatise we have

seen a very lovely interplay between geometry and algebra. Many important problems are still unsolved and new questions have been raised. I shall list some of them here.

1. Find new FMD’s of rank at least four.

A. Can SQ£*s and other known t - (v,k,l) designs be "erected"

to PMD's of rank greater than t + 1 ?

B. Do there exist "generalized spreads," as discussed in §4

of Chapter I, in the classical projective and affine

geometries? 2 6 2

C. Can rank 4 PMD’s be "erected" from BIBD's with \ = 1

by assuming additional regularity properties (for example,

the attempt by Dehon discussed in §6 of Chapter I)? Note

that in light of Theorem 1.4.20, constructing PMD's with

large independence numbers could be as difficult as

constructing t-designs for large t .

2. Find more interesting families of subloops of the

Bruck loop fl and the Malbos loop M . Can one construct an infinite cexp-3 loop which contains for each d £ N the free c-ty exp-3 loop of dimension d ?

3. Is the Triple Argument Hypothesis valid in all c-7)\ exp-3 loops?

4. What is the analogue of Theorem 4 .2.5 for finitely generated c—77? exp-3 loops of nilpotence class 3 (and greater than 3)? One must take into account the dependencies among the basic associators and the constraints they produce: recall

Theorem 5*2.5 and Proposition 6.2.2 .

5 . Is there a unique factorization of any finitely generated c-JT? exp-3 loop into irreducible loops as defined in Definition

4.3.2 ?

6. Can more identities involving dependencies among associators be proved using the method of Proposition 5.2.4 and

Proposition 6 .2.3 ? 263

7* Are the local designs of HTS's new in the sense that they can not he constructed using the methods discussed in [17] ?

8 . What is the precise structure of the Fischer groups and the full automorphism groups of HTS1s ?

In my mind, the most important and interesting problem is the first. I hope that the HTS's - both those constructed by others previously and those constructed here for the first time - will someday be known as the first of many non-classical higher rank

PMD's, all of which will surely be very beautiful geometries. BIBLIOGRAPHY

1. E. Bannai, On Tight Designs, Quarterly Journal of Mathematics, Oxford (2), Volume 28 (1977), ^33-448.

2. L. Beneteau, Topics about Moufang Loops and Hall Triple Systems, to appear in "Simon Stevin.

3. K. Biggs, Finite Groups of Automorphisms, London Mathematical Society Lecture Note Series (6), Cambridge University Press, 1971. G. Bol, Gewebe und Gruppen, Mathematische Annalen, Volume Ilk (1937), klCkil.

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