Nilpotent Algebras with Maximal Class
in Congruence Modular Varieties
by
Xuebin Zhang
A thesis submitted to the FACULTY OF GRADUATE STUDIES of the UNIVERSITY OF MANITOBA in partial fulfillment of the requirements of the drgree of
Doctor of Philosophy
Departme n t of Mathematics and Asuonomy University of Manitoba Winnipeg. Manitoba
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1 am deeply grateful to my supervisor Professor Robert Quackenbush. He awakened in me an interest in Universal AIgebra; He suggested to me to investigate quatemary Mendelsohn quasigroups, and this investigation has fimdly Ied to this thesis. He patiently discussed with me my ideas, and provided me with his rxperience.
1 would like to thank the Department of Mathematics and Astronomy, especidiy Professor Lynn Battm and Professor Nathan Mendelsohn for their support and encouragement
1 appreciate the tinancial support given to me by the Faculty of Gnduate Studies of the University of Manitoba, by the Departmant of Mathematics and Asuonomy and by Professor Robert Quackenbush. A bstract
Generai representation theory by polynomials for nilpotent algebns in modular congruence varietirs has been established and studied by sevenl authors. An intereshg open question is how to recursively constmct nilpotent aigebns with maximal class. In this thesis. severai algebras are investigated: quatemary Mendelsohn quasigroups. Steiner quasigroups. Steiner loops. Steiner skeins. and p-groups.
First. we obtain the structure theorem of finite nilpotent quaternary Mendelsohn quasigroups.
By recursive construction we prove that for any natural number n. (i) there exist a nilpotent quaternary Mendelsohn quasigroup of order 4"+' with maximal class n. and a solvahk quavmary Mendelsohn quasigroup of ordrr 4" with maximal class n;
(ii) there exist a nilpornt Steiner quasigroup of ordrr 3"+l with maximal class n. and a nilpotent Steiner loop of order 2n+2with maximal class n; (iii) any subdirectly irreducible quaternary Mendelsohn quasigroup (Steiner quasigroup) of class n+ 1 with some conditions cm be expanded to a subdirectly irreducible quaternary Mendelsohn quasigroup (Steiner quasigroup) of class n+2.
We aLso give a new and simple recursive construction for a nilpotent Steiner skeins of order 2n+2,th maximal class n such that di its denved Steiner loops are of nilpotence ciass n; and we tepment by polynomials the dihedral group and the generalized quaternion group of order 2"+' which are nilpoient p-groups with maximal class n. Table of Contents
Chapter 1. Introduction
A generai representation theory for nilpotent algebras in modular congruence varieties was established by Freese and McKenzie [7]. By applyinp this representation theory to some special algehns based on vector spaces, Guelzow [Z] obtained some generd representation theories by polynomials. Guelzow also posed an interesthg question, that is, are there any recursive constructions for finitr. nilpotent algehras with maximal class?
In this thesis. several aigebras ;ue investigated: quatemary Mendelsohn quasigroups, Steiner quasigroups. Steiner loops, Steiner skeins and p-groups.
In Chapter 2. we introduce basic concepts from group theory to universal algebn. and the gened representation theory of F~eseand McKenzie.
In Chapter 3, we smdy quatemary Mendelsohn quasigroups. In 197 1. Mendelsohn generalized Steiner triple systems to cyclic triple systems; later such cyclic triple systems were called Mendelsohn triple systems. Similar to Steiner quasigmups which are obtained from the CO-ordinatizationof Steiner triple systems. Mendelsohn quasigroups are obtained from the CO-ordinatization of Mendelsohn triple sysams. Recently, Quackenbush introduced quatemary Mendelsohn triple systems and their corresponding quatemary Mendelsohn quasigroups. We study some algbnic pmprrties with universal algeebra as a tool; we obtain the stnichire theorem of €mite nilpotent quatemary Mendelsohn quasigroups ; and hy recursive construction we prove that for any natural numher n:
(i) thex exists a nilpotent quatemary Mendelsohn quasigroup of order 4n+1 with maximal class n; (ii) there exists a solvable quatemary Mendelsohn quasigroup of order 4" with maximal class n; (üi) any subdirectly irreducible nilpotent quatemary Mendelsohn quasigroup of class n+l with some conditions can be expanded to a subdirectly irreducible nilpotent quatemary Mendelsohn quasigroup of class n+2.
In Chapter 4. we investigate recursive constructions of nilpotent Steiner quasigroups and nilpotent Steiner loops. Guelzow presented a strengthened version of the representation theorem given by Klosxk [ 151 for fmite distributive Steiner quasigroups. and generalized the theorern to the class of dl finite nilpotent Steiner quasigroups and posed the following open questions:
(i) Are there any recursivt: constructions for nilpotent Steiner quasigroups that raise the nilpatence class?
(ü) Are the= any such constmctions for distributive Steiner quasigroups?
For the fitquestion we give a recursive construction; the second question is su1 open. A similar result is given for Steiner loops. That is.
(i) There exists a nilpotent Steiner quasigroup of order Y+' with maximal class n.
(ii) There axists a nilpotent Steiner loop of order 2n+2with maximal class n.
(iii) Any subdirectly irreducible nilpotent Steiner quasigroup of class n+ 1 can expanded to a subdirectly irreducible nilpotent Steiner quasigroup of class n+2.
In Chapar 5. we give a recursive construction for a nilpotent Steiner skein of order
2n+2with maximal class n such that dl its derived Steiner loops are of nilpotence class n. Armanious and Guelzow [1,3] obtained the structure theorem of finite nilpotent Steiner skeins. Guelzow [3] gave a construction of a Steiner skein of nilpotence class n with ail derived Steiner loops of nilpotence class 1. Armanious [ 1J gave a construction for Steiner skeins of nilpotence class n with ai1 its hrived Steiner loops of nilpotence class n. In this chapter we survey the main mdts on nilpotent Steiner skrins and give a new and simple construction. in the form of polynomials. for Steiner skrins of nilpotence class n with all its derived Steiner loops of nilpotence class n.
In Chapter 6. we investigate recursive constructions of finite p-groups. Some general representation throrems for finite nilpotent goups and some examples for pgroups with small order have been given by Guelzow. By ~cursivi:construction we shail represent the dihedral group and the genenlized quaternion group of order 2"+' which are nilpotent pgroups with maximal class n. At the same time. we xe that the representation theories by polynomials provide a powerful tool for isomorphism classification of finite pgroups. Chapter 2. Preliminaries
1. Groups
In the theory of groups, the important concepts of Atelian (commutative) group. the center of a group, the centralizer of a normal subgroup, solvable group, and nilpotent group cmal1 be defined in tems of the commutator opention. that is, [x, y] = x'ly-'xy.
Altematively. these concepts can be defined in terms of the operation [M, NJ = Sg(([x. y]
: x E M, y E NI), on normal subgroups, also called the commutator (SEO()is the sub- group genented by X). Analogous conceprs. based on the multiplication of ideals, are important in ring thzory. An extension of these concepts to algebras other than groups and rings is behind one of the most exciting new directions of rexarch in pneral algebn.
The following detinitions and results can be found in [2 1,221.
Definition 1.1. An algcbra G = ( G; 0, -'.e ) is a group if it satisfies the following identities : (i) (xmy)ez = x.(y.z);
(ii) xe = e.x = x;
(iii) x.x-' = x -1 ex = e. A group G is called abelian if the commutative identity holds: (iv) x.y = y.x A group G is called qckc if G is generated hy one element.
Definition 1.2. A subgroup N of a group G is normal, if g~g-'= N for every g E G. Definition 1.3. If a, b E G, the commutator of a and b, denoted by [a, b], is
[a. b] = a%"nb.
Definition 1.4. The commutator subgroup (or derived subgroup ) of G. denoted by G'. is the subgmup of G genented by al1 the cornmutators. The higher commutator subgroups of G are defined inductively:
@O' = G, ~(i+"= ~(i)'.* that is. G('+')is the comrnutator subgroup of G?
Definition 1.5. A group G is solvable if G'")= {e) for some n. and the least such n is caed the class of the solvable group G. Let H. K be normal subgroups of G; de fine
[H.KI = Sg({ [h. k] : h E H and k E K )), which is a normal subgroup of G.
Definition 1.6. Define the subgroups vi(G) of G by induction:
w@) = G. vi+(G) = [vi(G). Gl* The lower central suries of G is the series
G=yo(G) Ly!'(G) 2.- - -
Defînition 1.7. The centrr of a group, Z(G) = ( a E G : asx = xsa for dl x E G }. and note that it is a normal subgroup of G. The upper central xriss { ci(G)) is the series of subgroups of G defined by induction:
co(G) = {el. (a) = co(G) S ci(G) S cz(G) 5 . . . . Definition 1.8. A group G is nilpotent if y,(G) = { e } for some n. and the least such n is cailed the c[crrs of the nilpotent group G. Theorem 1.9. If G is a group. then there is an integer n with I;,(G) = G if and only if vn(G) = {el. Monover. in this case. vi(G) I <,JG) for al1 i. Theorern 1.10. G is nilpotent of class n > 1 if and only if GIZ(G) is nilpotent of class n- 1. 2. The commutator and the center General commutator theory has to do with a binary opention. the commutator. that can be defmed on the lattice of congruences of an algebn. The opntion is very well behaved in conpence modular varieties. but much Iess so in most other vuieties. In 1976. Smith in his book on Mal'cev varieties [24] genrdized the group theoretic concept of the commutator to the theory of Mal'csv vxieties. With this concept he also generalized such notions as the 'center' and 'nilpotence'. In 1979, Hagemann and Henmann [13] extendeci the theory of the commutator to modular varieties. In 1980 and 1983 Gurnm [9. 101 presented an introduction to commutator theory which motivates the commutator geometricalIy. In 1987, F~seand McKenzie [7] published a book on commutator theory for congruence modular vzuieties, a more complete introduction to commutator theory. The following definitions and results cmk found in [7.9. 10. 161. Definition 2.1. Let A be any algebra. The center of A is the binary relation c(A) defined by : ca. b s E c(A) if and only if for every n 5 1. and for every terrn opention t E C~O~+~A. and for dl( cl. c2. -.. . c,) and (dl?d2. ... . d,) E An, t(a,c,. ..-.cn) = t(a. di, ... .dn) t, t(b.cl. ... ,c,) = t( h. dl, ... ,do). A is called abeiian if and only if c(A) = AXA. Definition 2.2. Let a,P, q be congruences of an algrhra A. We say that a centralizes 13 modulo q . writtrn p, q). if and only if for ewry n 1 1. and for every am opention t E CIO.+~A. and t(&ci..-. .cd 3 t(a,di ..... d,) t, t(b.ci ..... c,) qt(b.di..... d,). Definition 2.3. For congruancas a and B of A. wz defini: their cornmututor, denoted [a,pl, to be the smallest congruence q of A for which a cenrralizes P modulo q. The centralizer of modulo a denoted (a: P). is the largest congruence y of A such that y cenvalizes modulo a. Lemma 2.4. Let OA and 1A hc: the lest and largest congruences of A. Then (i) c(A) = (OA : LA). that is. the largest congruence a such that [a. 1A] = OA; (ii) [LA, lA]is the srnailest congruence on A such that A/[lA. lA] is abdian; in particular, [ 1A. 1A] = OA iff c(A) = lA iff A is aklian; (iii) [a, 5 a n P. The commutator has proved to be a powerful tool for investigating conguence modular varieties. This is Iargely due to the fact that for any dgebra in a conpence modular variety. the commutator is a commutative and completdy join-presewing operation on congruences. We nfrr the reader to [7] for an extensive discussion on modular varieties and the commutator in thest: vacieties. Theorem 25. (Mal'cev [7]). A variety V has permuting congruences iff there is a 3-ary term p such that the equations p(x, y, y) = x. p(x. x. y) = y are valid in V. Theorem 2.6. (Jonsson [7]). A variety V has distributive congruences iff there exists an n and 3-;isr tems do. . . . . d, such that the following equations hold in V. (i) do(x. y. z) = x; (ii) di(x, y, x) = x tor 0 l i 5 n; (iii) di(x. y. y) = di+l(~.y, y) for al1 even i < n; (iv) di(x, X, y) = di+l(~,X, y) for ail odd i c n; (v) ddx. y. 2) = z- Theorem 2.7. (Gumm [7]). A variety V has modulcir congruences iff there exists an n and Zary terms p and do. . . . . dn such that the following equations hold in V. (il &(x. y, 2) = x; (ii) di(x. y. x) = x for 0 5 i 5 n; (iii) di(x. y. y) = di+i(~.y. y) for al1 even i < n; (iv) di(x. X. y) = di+i(~,X. y) For al1 odd i < n: (VI ddx. y. y) = p(X. y. y); hi) p(x, x. y) = y. Definition 2.9. Let a and P be congruences on an algebra A. A$ is the congruence generated by { ( x x) y y) ) : x P y } in a, where a is taken as a subalgebra of AXA. Lemma 2.10. (Gumm [9]). If A is an algebra in a congnirnce modular variety then the commutator [ ci, p ] = { (d. c) : (b, b) A~S(d. c) }. Definition 2.11. Let V be any variety. A temary tenn d(x. y. z) is called a Gumm difference term if it satisfies the following two conditions: (i) d(x, x. y) = y is an identity in V. (ii) If (x. y) E 6 E Con A. for some A E V. then d(x. y. y) [&O] x. It is well known thai every modular variety has a Gumm difference term and that in a permutable variety the Mal'cev krm is a Gumm differenca tem. =d( f( rda.b). .-.. r.(a.b)), f( rl(b.b). .... rn(b,b)), f( cl. ... .c,) ). and (ü> d( r(a,b). r(b.b). r(b,b) ) = r(a.b), for al1 f c r, al1 ( cl. ... . cn) E An ( n being the anty off ) and for al1 binary term functions rl(x. y). . . . , r,(x, y). Corollary 2.13. Let c A, T > be an algebn in a permutable variety and let p(x. y. 2) be a Mai'cev tetm. Then a c(A) b if and only if f ( p( rda.b). rl (b.h). cl 1. ... . p( r.(a,b). r,(b.b). cn = p( f ( rl(a.b). ... . rn(a.b)). f ( rl(h.b). .... r,(b.b)), f ( ci. ... . c, ) ). for al1 f P r, a11 ( cl. ... . C, ) E An ( n being the arity off ) and for dl binary term functions rl(x. y). . . . . rn(x, y). 3. Abelian and affine algebras Definition 3.1. Let < A : F > be any algchra. A is called amne if there exists an abdian group < A ; +, -, O > and a temary km function r(x. y. 2) of A such that (i) r(x, y, z) = x - y + z for al1 x, y, z E A, and for each n-ary operation f E F there are endomorphisms ai... - , a, of the abelian group < A : +, -, O > and c E A such that (ii) f(xi. - . - . xn) = ai(xl) + . . . + a&,) + c. If this abelian group exisis it is called the group arsociated with A and the term function T is cdled a difference fincîiun for A. Theorem 3.2. (Hemnann [12]). The following are equivalent for an aigebra A in a modular variety : (i) A is abelian; (ii) A is affine. 4. Nilpotence and solvability The foliowing definition cmhe found in l2.71. Definition 4.1. Let A be an algehra. Define [lAl0=lA. [lAll= [ IA,lA], [L~]~~~= [ [lAlk,[lAlk ] for k 2 1. Definition 4.2. An algahra A is soivuble if [lAIn= OA for some n. and the least such n is called the class of the solvable algebra A. Detinition 4.3. De f ine vo(A) = 1~. W+~(A)= [w(A). 1~1for i 2 0. The bwer central serius of A is the series G=vo(A) >vi(A) 2... Definition 4.4. An algebra A is nilpotent if y,(A) = OA for some n. and the least such n Ïs called the cluss of the nilpotent algebn A. A nilpotent aigehn A of class 1 is abeliun. Definition 4.5. Define co(A) = On L;i+l(A)Ici(A) = c(A/ci(A)) for i 0. The upper central series of A is the senes 0A=co(A)5 GI(A)S G2(A)S.. . - Theorem 4.6. Let A be an algebra in a modular vaiety. Then A is nilpotent of class k if and only if c, = I A and ck-i# 1A. Theorem 4.7. Let A br: an algehra in a modular variety. Then A is nilpotent of class k > 1 if and only if A/C(A) is nilpotent of class k- 1. 5. Representing nilpotent algebras The main tool in the proof of somi: representation theomms is the description of the stniciure of the non-affine algebras in a congruence modular vwïety given by Freese and McKenzie in 171. This description uses the following concept of a product of two algebns: Definition 5.1. (Freese and McKenzie [7]). Let < M; {Fi)iE1 > and < N; {FIisI> be algebras in a congruence modular variety. ktM tK: affine with associated group < M;+. -.O >. ktT ha a system of maps Ti: ~4 -t M. (for i E 1) where ti is the anty of P. Then A = N @T~= < NxM; {FiJi > is drfinçd to be the algebn with : ~'((n1, ml).. . . . (nt,, m,,))= ( Fi(n . . . . nt,). Fi@ 1. . . . . m,,) + Ti(n l, . . . . nti) ) where (ni, ml). . . . . (nt,*mq) E N x M. Theorem 5.2. (Freese and McKenzie [7]). Let A be an algebra in a congruence modular variety V. Let N = A/I;(A). Then there exists an abdian algebn M in V and asystem T of maps as described ûbove such that A s N @*M and the center of N @*M is the kernel of the projection ont0 N. Corollary 5.3. An algebra in a congruence modular variety V is nilpotent of class 2 or less if and only if it can be represented as (that is, is isornorphic io) N~@~N~where NIand N2 are abelian algehras in V. Corollary 5.4. If an algebra in a congruence modular variety V is nilpotent of class n. then it cm k represented as (that is, is isomorphic to) (( . . . ((N 18'' N2)@Tz N j) . . . ) @Tn-~Nn). where Ni. . . . N, art: aklian dpbras in V and Tl. . . . . T,., are some systems of maps as described above. 6. Subdirectly irreducible algebras The following definitions and results cm be found in [8]. Definition 6.1. An algebra is callad subdirectly irreduciblr if for any subset C c Con(A), the relation n(a~C)=OA implies the existence of B E C such that P = OA . Corollary 6.2. An dgebra A is subdirectly irreducible if and only if it has only one element or Con(A) has one and only one atom. which is contaïneci in every congnience relation other than O*. 'Iheorern 6.3. (Birkhoff [BI). Every algebra is isomorpic to a subdirect product of subdirectly irreduci ble algehras. The following bmma is an exercise in [7]. Lemma 6.4. Let A be a nilpotent algebra in a congruence modular variety. Let a E Con( A). Then a # OA if and only if a n c(A) # OA. Lemma 6.5. A nilpotent alpbra in a congruence modular variety is subdirectly irreducible if and only if for any subset C c { a E Con(A) : a S c(A) } the relation n(a~C)=OA implies the existence of E C such that P = OA. Chapter 3. Quaternary Mendelsohn Quasigroups 7. Quaternary Mendelsohn quasigroups Definition 7.1. A cnrnbinatorial quasigroup is an ordered pair (A, O). where A is set and "O " is a binary operation on A such that for dl not necessarily distinct a. b E A. the equations aox = b and yoa = b have unique solutions. Definition 7.2. An algebraic quosigroup is an algehra (A; O, \. 1). such that " O ". " \ ", " 1 " are three binary operations on A saiisfying the four identities: (i) xo(x\y) = y; (ii) x\(xoy) = y; (iii) (x/y)oy = x; (iv) (x0y)Iy = x. Supposr (A. 0) is a cornhinatonal quasigroup. We cm define: two binary operations " \". "1" onA by a&=c iff a\c=b iff &=a. Itisnotdifficulttoseethat (A; 0, \. /) is an algebraic quasigroup. On the other hand. if (A; O, . is an algebraic quasigroup then each of (A. O). (A. \) and (A, 1) is a combinatorid quasigroup. In 197 1. N. S. Mendelsohn [ 171 genrralized Steiner triple systems to cyclic uiple systems; later such cyclic triple systems were called Mendelsohn triple systems [27,28]. Similar to Steiner quasigroups which are obtained from the CO-ordinatization of Steiner triple systems. Mendelsohn quasigroups are obtained from the CO-ordinatization of Mendelsohn triple systems. Definition 7.3. A Mendelsohn triple system is a pair (X. B). where X is a set of elements (called points) and B is a collection of cycles with size three such that each ordered pair of distinct points of X appears in exactly one cycle of B. (The ordered pairs (a. b). (b. c). (c. a) appear in the cycle (a. b, c) and the cycles (a. b. c). (b. c. a). (c. a b) are al1 equal. ) Definition 7.4. A Mendelsohn quasigroup is a corn hinatonal quasigroup (A. 0) satisfying the following identities: (i) xox =x; (ii) (XOY)OX = y. Lernma 7.5. If (X,B) is n Mendelsohn triple system. define A = X. aoa = a for al1 a E A and otherwix aob = c if and only if (a. b. c) E B. Then (A, O) is a Mendelsohn quasigroup. If (A. O) is a Mendelsohn quasigroup. dafine X = A and B = { (a, b, C) 1 a # b and aoh = c ). Then (X. B) is a Mendelsohn triple system. Lemma 7.6. An algebn (A. 0) is a Mendelsohn quasigroup if and only if (A. 0) satisfis the following iden ti ties: (i) xox = x; (ü) (x0y)ox = y. Recently. R. W. Quackenbush introduced quatemary Mendelsohn triple systems corresponding quatemary Mendelsohn quasigroups. Definition 7.7. (Quackenbush). A Mendelsohn tripIl: system is quaternary if for any cycle (a, b. c) thex is a unique fourth point d such that (b. a. d). (c. b. d) and (a. c. d) are cycks. Definition 7.8. (Quackenbush). A Mendelsohn quasigroup (A. O) is quatemary if it satisfks the identity : (iii) (xoy}oy = yox. Lemma 7.9. An algehn (A. O) is a quatemary Mendelsohn quasigroup if and only if (A. O) satisfizs the following idantities: (i) xox =x; (ii) (x0y)ox = y; (üi) (x0y)oy = yox. Lemma 7.10. A quatemary Mendelsohn quasigroup satistks the following identities : (iv) XO(~OX)= y; (v) xo(x0y) = yox; (vi) (xoy)o(yox) = x. From the fact that quatemary Mendelsohn quasigroups are algebraic. we have that the variety of quatemary Mendelsohn quasigroups is a congruence uniform. regular. coherent. permutable and modular vdety. That is. Lemma 7.11. The variety of quaternary Mendelsohn quasigroups is a modular variety. Proof: Since the= is a Md'cev ûxm p(x. y. Z) = (xoy)o(zox) with p(x. x. y) = y and p(x, y, y) = x. it is congruence-permuiable and so is congniencr-modular. Lemma 7.12. Let c A; O s be a quatemary Mendelsohn quasigroup. and a a congruence of A. Then (i) al1 congruence clases of a are of same six; (ii) each congruence class of a generates the congruence a; (iii) each congruence clüss of a is a subalgehra; (iv) each subalgebra that contains a class of a is the union of congnience classes of a. Lemma 7.13. If an idempotent combinatorid quasigroup satisfias the medial law. that is, (xoy)o(zot) = (xoz)o(yot). then (i) SO(XOZ)= vo(xot) implies so(yoz) = vo(yot); (ii) (x0z)on = (x0t)orn implies (y0z)on = (y0t)om. Proof: Let x = you. Since { so(xoz) } 0 { vo(x0t) ) = (s0v)o { xo(z0t) } = (sov)o{ (you)o(zot) } = {so(yoz) } o{vo(uot) } and vo(x0t) = vo{ (you)ot } = {vo(yot) }O (vo(u0t) J. So so(yoz) = vo(yot) and we have (i). Similady, we have (ii). Lernrna 7.14. The following are quivalent for a quaternary Mendelsohn quasigroup < A: 0 > : (i) c A; O > is affine; (ii) < A; O > is abelian; (üi) c A; O > satisfes the medid law, that is. (xoy)o(zot) = (~o~)o(yot). Proof: From Lemma 7.1 1 we have the variety of guaamary Mendelsohn quasigroups is a modular variety. and then from Theorem 3.2, (i) and (ii) are equivalent Assume it is abelian; since (z~y)o(zoz)= (zoz)o(y~z), we have (zoy)~(zot)= (zoz)o(yot). and then (xoy)o(zot) = (xoz)o(yot). Assume it satisties the medial law; then it is abelian from Lemma 7.13. Definition 7.15. Let < B; O > be a subdgebra of c A; O >. c B; o > is normal if B is a coset of a congruence on c A: O >. Lemma 7.16. Let < A: O > he a finite quatemary Mendelsohn quasigroup. If it satisfies the medial law, then IAl = 4" for some n 2 0. Proof: Sincr it saisfies the medial law. we have (x. y. xoy, yox } is a nomial subalgebn for any x, y E A. Lemma 7.17. Let < A: O z he a quatemary Mendelsohn quasigroup. Then each class of the center is a subalgebn which satisks the medial law. Proufi This follows from Definition 2.1 of the center. Theorem 7.18. Let Proof: This follows from Lemma 7.14, Lemma 7.16 and Theorem 4.7. Proof. Apply Comllary 2.13 with Mal'cçv term p(x. y, z) = (xoy)o(zox) and note that the only binary term functions are x. y. xoy. yox. 8. Quaternary Mendelsohn quasigroups of order 4 and 4' Let a k a congruence with class six4 on a quatemary Mendelsohn quasigroup < A, 0 >. First we ask whather it is true that a 5 c(A); we shall show the answer is negative. In this section and the Sections 9 and 10 we always let GF(4) = (0. 1.8. 1+ 0) with e2= 1 +8. From Lemma 6.5 we have Lemma 8.1. A nilpotent quatemary Mendelsohn quasigroup with the size of its center classes king 4 is subdirectly irreducible. Lemma 8.2. Define aob = Qa + (1-1-8)b for a. b E GF(4). Then < GF(4), l > is 20 an abelian quatemary Mendelsohn quasigroup which is simple and henct: subdirectly irreducible. Lemma 8.3. Let A he a quaternary Mendelsohn quasigroup with 4 elements. Then A is isomorphic to < GF(4), >. Notice that if we define a*b = (1+0 )a + 8b for a, b E GF(4). then cGF(4), *>is a 4element quatematy Mendelsohn quasigroup. and hence isomorphic to < GF(4), >. It is the opposite of < GF(4), > since x*y = y.x. Clearly. on any 4-ebment set there are exactly two ways of defining a quatemary Mendelsohn quasigroup and these are opposites of each other. Cal1 any 4-elrment suhquasigroup of a quatemary Mendelsohn quasigroup a quanet. If A is any quatemctry Mendelsohn qusigroup containing the quartet Q and we replace Q with its oppositc. then the modified dgrhri is again a quatemary Mendelsohn quasigrou p. Theorem 8.4. Let < B. > and c H, l > he two quatemary Mendelsohn quasigroups of size 4 where B = { a. b. ah. ha) and H = { c. d, cod. dec ). kt A = ( (u. f) : u c B, fa H }. Define "O " on A as hllows. (u. &(Y. g) = (U.V. pf) if u = v = bma ; (i) < A, O > is obtained from c B, > x < H, > by replacing the quartet {(boa. h) I h E H) with its opposite and consequently is a quatrmary Mcndelsahn quasigroup: (ii) q = { < (u. 0, (u, g) s : u E B, f, g E H ) is the unique atomic congruence, which 21 is contained in avery congruence relation other than OA rnaking < A, 0 > subdirectly (üi) < A, O > is not nilpotent; (iv) < A, O > is solvable of class 2; Proof (ii). First it is ~adilychecked that Y) is a congmence. To see this. note that if a is a congruence of the quaternary Mendelsohn quasigroup M and M' is a subquasigroup contained in some hlock of a, and MW is a quatemary Mendelsohn quasigroup on the same set as M'. then M* = (M - M') u M" is a quatemary Mendelsohn quasigroup and a is a congruence of M*. Since itç class size is 4, it is an atom. In the following we show that it is the only atorn. Let 0 be a congruence other than OA and q; then thex are two slements, say, (u. fl and (v. g), with u * v. u z bea and (u. f) 9 (v, g). If f* g. then x 8 y for x. y E El = { (u, f). (v, g). (U.V. bg). (W. gof) }. By multiplying Elon the left hy (u, g), (u. fq), (u. g4). respactively. we have Ez = ( (u. pf), (v. f@. (U.V. p), (v~0 1, % = { (u*g). (v. 0. (U.V. g.0. (v.u, f*g) 1, E, = { (UV f.g). (v. g4. (U.V. 0, (v.u. p) 1; and we see that x 8 y for x, y E Ei where i = 2. 3,4. Wr let v = boa without loss (u, g) 8 (u, Lg), implying T( < 8. If f = g. then takuig h + f. then we see that x 0 y for x. y E Fi, where 1 li 14 and F1={(u.f). (V.O. (U.V. 0, (v*u, f? 1, FI ={ (u. h*Q. (v. h4), (U.V. h*f), (vmu. h*t) }, F3 =( W. h). (v, h). (U.V. h). (v.u, h) }, F4 ={ (u. fmh). (v. fmh), (U.V. fah). (WU.feh) }. We let v = boa without loss generality, than (u. Qo(u. h.0 0 (v. oo(v, h.0. that is, (iii) and (iv). In the following we apply Definition 2.9 and Lemma 2.10 to show ( (boa. od). (ha. c.d) ) &p ( (a. d.c). (a. d) ). so (a. dc) [ cc, P (a. d); that is. (a. de) [ 1A, 1~1(a, dl. From this. we have [lA, LA] = { < (u, t), (u, g)>: u E B, f, gc H ). (bma. c) [ a, $ ] (boa. cod); that is. (hma, C) [LA, [l~.lA] ] (boa, cad), and then [lA,[ IA, lAJ j = [ IA,lA]. Tharefore. it is not nilpotent by the definition, and it is easy to ser that [ [ lA. Id. [lA, LA] ] = OA, so it is solvable of class 2. (v). This follows tiom (iii). The lattice of al1 congruences on < A, 0 > 9. Representing nilpotent quaternary Mendelsohn quasigroups Applied to quaternary Mendelsohn quasigroups the descriptions in Definition 5.1. Theorem 5.2, Corollary 5.3 and CorolIary 5.4 therefore becorne : Definition 9.1. Let < M; O > and c N; O > k quaternary Mendelsohn quasigroups. Let M be affine with associated group c M; +. -, O >. Let p be a map from NX N -+ M. Then A = N Q'M is defined to be the algebn on Nx M with : Theorem 9.2. Let < A ; O > be a quatemary Mendelsohn quasigroup. Let N = AII;(A). Then there exists an abelian quatemary Mendelsohn quasigroup M and a map p as describeci above such thût A = N d M and the center of N €3'~is the kernel of the projection onto N. Corollary 9.3. A quatemary Mendelsohn quasigroup is nilpotent of class 2 or less if and only if it can ht: reprexnted as (bat is, is isornorphic to) N N2 where Ni and N2 are abelian quatcrnary Mendelsohn quasigroups. Corollary 9.4. If a quaternary Mendelsohn quasigroup is nilpotent of class n, then it cm be reprwntrd as (that is. is isomorphic to) where Ni, +.- N. are ahelian quatemary Mendelsohn quasigroups. and pi. . . . . p,.l are some systems of maps as described above. Since every aklian (that is. medial) quatemary Mendelsohn quasigroup is isomorphic to < (GF(4))"; > with xmy = ex + (l+e)y. Theorem 9.2 cm be used to prove the following reprwnriition thttorem by induction over the class of nilpocnce : Theorem 9.5. Let A = < A; O > be a finite quatemary Mendelsohn quasigroup of nilpotence class k. Then there exists an rn-dimensional vector space R, a polynomial p: Q2 + R over GF(4). and a sequence 1 6 ni < .. . < nk = m of integers such that (i) if 1 d s < k and n, < i In,,~, then pi(x, y) dors not depend on the variables X*+l. --• . X, and Y",+l. **. Ym; (ii) pi(x. y) = 0 for al1 x, y c R and 1 li ln 1; (ci) = < R; O > is isomorphic to A where xoy = ex + (8+ l)y + p(x. y); (iv) the crnter of A corresponds to the kemel of the projection on to the first nr.1 components of Q. This projection is a homomorphism. From Theorem 9.5. we have the following theorem. Theorem 9.6. Let < A: O > bt: a tinite quaternary Mendelsohn quasigroup of order 4? Then < A; O > is nilpotent if and only if it is isomorphic to < Q: O >, where R is an for al1 x, y E A. Proof: Wr only need to show (iii). Since xox = x if and only if p(x. x) = 0. (xoy)ox = y if and only if Bp(x. y) = p(xoy. x). and (xoy)oy = yox if and only if Lemma 9.7. Let A be an rn-dimensional vector space over GF(4). Define c A ; O > is a quatemary Mendelsohn quasigroup. Thrn wr have (i) p(x. x) = O for al1 x; (ii) 8p(x. y) = p(xoy, x) for al1 x. y E A; (iii) 0p(x, y) = p(xoy. y) + p(y. x) for al1 x. y e A. Fmn (ii) and (iïi) we have, for al1 x, y E A, P(X. Y) = p(x. Y). pWy. x) = 0p(x. y). p(y. xoy) = eZP(x, y); p(y, x) = p(y. XI. p(y0x. y) = Q(y. x). p(x. yox) = 82p(y.x); pboy. Y) = 8p(x. y) + p(y. x). p(y0x. xoy) = e2p(x. y) + 8p(y. x). From Theoran 9.6 and Lemma 9.7 we have Theorem 9.8. (Nilpotent Quatemary Mendelsohn Quasiproup Construction) Let < A ; 0 > be a nilpotent quatemary Mendelsohn quasigroup of order 4m. Define R = AX GF(4) and (x. i)o(y. j) = (xoy. 0i + (l+e)j + p,+i(x. y) ). Then c Q ; O > is a nilpotent quaternary Mendelsohn quasigmup if and only if the following identities hold: (i) P,+~ (x. X) = O for ail x E A, (ii) @p,+l(x, y) = p,+l(xoy. x) for x. y E A, Moreover, this is tmt: if and only if for any Csubdpbra B = ( x. y. xoy. yox } of A, Theorem 9.9. A finite quatemary Mendelsohn quasigroup is nilpotent if and only if it is isomorphic to a quatemary Mendelsohn quasigroup obtained from the klement quatemary MendaIsohn quasigroup by repeated application of Nilpotent Quatemary Mendelsohn Quasigroup Construction. 10. Recursive construction for nilpotent quaternary Mendelsohn quasigroups with maximal class In this section we detinr ah = 8a + (1 +9 )b for a. b E GF(4) and (XOY)~= Xieyi = exi + (l+0 )yi for X. y E GF(4)". Lemma 10.1. Let c A ; O > be a quatemary Mrndrlsohn quasigroup and a,b E A witha#b. Let H={a, b, aoh, boa}. x E A with xé H and B = {aox,a, x,xo~). If b> E c(A). then E = (mou : m E B and u E H) is a sub-quatemary Mendelsohn quasigroup which is isomorphic to the direct product of two Celement quaternary Mendelsohn quasigroups B and H. Proof. It is easily checked thrit (rnoa)o(noa) = [(mon)oaJo[(mon)oaJ for m. n E B. Since CU, a> E c(A). we have (mou)o(nou) = [(rn~n)ou)]o[(mon)ou)~ for m. n E B and u E H. That is, (m~u)~(n~u)= (mon)o(uou) for m. n E B and u E H. Since CU. v> E c(A). we have (mou)o(nov) = (mon)o(uov) for m. n E B and u, v E H. so that it is a quatemary Mendelsohn quasigroup. Then pW. YI) = (x12 + y12) { (143 )(a+b)xi + Wa+b)yi + W+b) + a 1. where p(0. 1 ) = a and p( 1. O) = h; moreover. xoy satisfies the media1 law. That is. a nilpotent quatcmary Mendelsohn quasigroup of order 16 is abelian. Proof. From Lemma 10.1. we sec that < A ; O > is abelian. that is. a nilpotent quatemary Mendelsohn quasigroup of ordrr 16 is abalian. In the rollowing we show that Q = ( ~(0).~(1). v(( l+@. v(0) lT. From kmrna 9.7, we have v(0) = (O. a. û(l+B)+h. h(l+B)). v(1) = (b. O. a(l+0). a+b(l+8)). v(l+e) = ( a& aû+b. O. aû+h(I+B) ), v(0) = ( a+% hg, a(l+B)+bB, O). 29 we have AY = Q; and then, Y = A-IQ (A-~)T; that is, p(xl. y,) = (x12+ y12) { (1+0 )(a+b)xl + O(a+b)yl + 8(a+b) + a }. Lemma 10.3. Let A be a 3-dimensional vector space over GF(4). Let xoy = €lx + (l+Q)y+ p(x. y) and pl(x. y) = pz(x, y) = O. If (a) < A: O z is quatemary Mendclsohn quasigroup for cases (1). (2) and (3); (b) c A: O > is ahelian in cases (1) and (2); (c) < A: o > is nilpotent of class 2 in case (3); (d) É A: o s is subdirectly irreducihle in case (3). It i.also readily checked that For cases ( 1) and (2) the medial law holds, so (b) holds. For case (3). tïrst note that it is not abelian since xo(y0z) # (xoy)o(xoz) for x = (0.0.1). y = (1.0.0) and t = (O. 1. O). By the results of the section 5. we know that A is nilpotent of class k with 2 5 k < 3 and that A/L;(A) is nilpotent of class k-1. Since I A/L;(A) I 5 16, it is of class 1. and so k = 2 and (c) holds. But a = { < x. y > : xi = yl, xz = y2 } I; c(A) and A/a is of class 1 as I Na I = 16. This forces = c(A) and (d) holds. Theorem 10.4. Let A he an (m+l)-dimensional vector space over GF(4) with xoy = ex + (1+0)y + p(x. y). If we define p(x, y) as follows: (il pdx. y) = pz(x. y) = 0; (ii) p&. y) = (xk-, + yk-1 + p~-~(x,y))(xly~ + ~2~1)for 3 5 k 5 m+l. then e A; 0 > is a nilpotent subdirectly irreducible quatemary Mendelsohn quasigroup of class m. Proof. By induction. it is readily checked that for 1 S i 5 m+l. (iii) 8p(x, y) = p(xoy, y) + p(y. x) for ail x. y E A. Thus. c A; O > is a quatemary Mendelsohn quasigroup. To prove < A; 0 > is nilpotent of class m. it is sufficient to show that c(A) = a ..-( - iff bi = O for 1 I i I m. By induction. Na is nilpotent of class m- L and hi = O for 1 5 i S m - 1. Thus, we assume that b = ( 0. . . . .O. b,. b,+[ ) and let b c(A) 0. By applying Lemma 7.19 with rl = O and r2 = b we have xoy = (oob)~{ { ( ( OO~)O( xoa) loy )o(~oo) j. From the dehition of xoy. we see that xoy = Bx + (l+B)y if either xi = x2 = O or and so we must have xoy = (b+x)oy + Ob. while (x~y),+~= ( (b+x)oy + Bb ),+l iff b, (xly2 + xzyi) = O. If we take xi = y* = 1 and x2 = y1 =O, this then forces b, = O. as was to be proved. Theorem 10.5. Let A be an m-dimensionai vector spacr over GF(4) and R = A x GF(4). Let (A: 0) is subdirectly inducible quatemary Mrndelsohn quasigroup of ciass n with XoY = ex + (l+@y + p(x. y). where (il pdx. y) = pz@. Y) = 0; (ii) pi(x. y) = pi(xl. .. .. Xi.1. Yi, . ... yi.1) for 2 5 i 5 m; (iii) p(x. y) = p(y. XI; (iv) p(x. y) =O if x =O or y = O. Define (x. &(y, j) = ( x~y,8i + (l+B)j + p,+i(x. y) 1. where class n+ 1. Proof. First it is rradiiy checked that so < R; O > is a quaternary Mendelsohn quasigroup. To prove c R; > is subdirectly irreducible of class n+l. we only need to show 33 that if b=(O..... O.b,.b,+i) and bc(R)O then b,=O. B y applying Lemma 7.19 with ri = O and rz = b we have xoy = (Oob)o{ { ( ( Oob)o( xoO) )oy } o(hoO) ), and we have and then and then 11. Recursive construction for solvable quaternary Mendelsohn quasigroups with maximal class Theorem 11.1. Let A = (GF(4))m with xoy = ex + (1+8 )y + p(x, y). If (i) pl(x, y) = 0. and (ü) forkl2. pk(x.y)=(xk+yt) for xi=yi=8 and i= 1.2. ... .k-l.and then ( A; O ) is a solvable subdirectly irreducible quatemary Mendelsohn quasigroup Proof: First it is readily checkcd that ( A; O ) is a solvabIr: quatemary Mendelsohn quasigroup. We take a = 1 f3 = 1 x = (O,O,O,O, ..., O), y = (8. 1, O. O, ... . O). u = (O. 1, O, O, ... , O), s = (0, O, O, O, ... . 0). v = (1. 1+8,0,0,.... O), t = (1+8.8,0,0,... 3). We have, (v. t )O( (y. y)o(u* s)} = ( (O, 0. 0. 0. ... , 0). (0. 1. O* 0. ... . 0)), and then, ((0, 1+8,0.0, ... ,O), (8. 1+8,0,0, ... ,O)) = ( (O. 0.0. 0. ... .O). (O, 1. 0. 0, ... . 0) ) (~~6). and then, (0.8. 0. 0,... . 0) [ a. B j (O. 1. 0. 0. ... . 0). so (0.0,O.O.... .O)[% ] (O.az.O.0.... .O) for a2c { 0. 1. 1+0.8 1. We take x = (O,O,O,O ,..., O), y = (8,8, 1. O, ... , O), u = (0,1,0,0 ,..., O), s = (8,8, O, O, ... . O), We have and then, (0. 1.8.0,...,O) [a,pl (O, 1. 1.0.... .O), and then, (O. az. O. O. ... . O) [ c* p 1 (o. a2. a3. 0. .-.. 0) Hence (0.0. 0. 0. ... . 0) [ a, ] (O. az, a3, a4. .-.. a,.l. a,) for a2. a3, ... . a, E ( O, 1. Ife. 8 ). Therefore ( A; 0 ) is a solvable quatemary Mendelsohn quasigroup of class m. Similar to the proof of Theo~rn8.4 we can show that it is subdirectly imeducible. The lattici: of al1 congruences on < A, O > Chapter 4. Co-ordinatizations of Steiner Triple Systems A Steiner triple system is a pair (X. B), where X is a set of chrnents (called points) and B is a collection of subsets of size three (called blocks) such that =ch pair of distinct points of X appws in exactly one block of B. Steiner quasigroups and Steiner loops aise from the CO-ordinatizationof Steiner triple systems (see i2.4, 1 1. 18, 19 1). Guelzow [2.4] p~srnteda strenphtened version of the representation throrem given by Klossek [IS] for tkite distributive Steiner quasigroups and genardized this theorem to the class of dl finite nilpotent Steiner quasigroups. He posed the following open questions: (i) Are the= any recursive constructions for nilpotent Steiner quasigroups that raise the nilpotence class ? (ü) An there any such constructions for disuibutive Steiner quasigroups ? For the tïrst question we will give a recursivt: constuction; the second question =mains open. Similar nsults will be givrn for Steiner loops. 12. Steiner quasigroups Definition 12.1. A Steiner quusigroup is an algebra which has a hinary operation sahfying the following identities: (i) xox = x; (ii) xoy = yox; (iii) (XOY)OX= y. A Steiner quasigmup is cailed distributive if it satisties the distributive law : (iv) (x0y)oz = (xoz)o(yoz). A Steiner quasigroup is called medial if it satisfies the medial law : (v) (xoy)o(zot) = (xoz)o(yot). Lemma 12.2. A Steiner quasigroup is a combinatorid quasigroup. Lemma 12.3. Let A be an m-dimensional vector space over GF(3). and p a polynomid: A~ -+ A aver GF(3). Detïnr xoy = - x - y + p(x. y). men A = < A : > is a Steiner quasigroup iff p(x. y) satisties: (i) p(x. x) = O for al1 x; (ii) p(x. y) = p(y. x) for dl x. y E A ; (iii) p(x. y) = ~(XOY,X) for dl x, y E A . Proof. xox = x iff P(X, X) = 0; xoy = YOX iff p(x. y) = p(y. XI; (XO~)Ox = y iff p(x. y) = p(x0y. x). Lernma 12.4. Let c A, > bç: a Steiner quasigroup. Then a ((A) b if and only if for for rl, r2 E {a, b. aob). That is, (i) xoy = ( ho{ ( ( aox)~b )o((aoy)ob) 1 )OX (ii) xoy = { bo{ ( ( aox)ob )oy 1 O (aob); (iii) x~y= ( ( aox)ob )O(((aOb)Oy)Ob); (iv) xoy = ( bo( xo(((aob)oy)ob) } los; (v) xoy = { bo{ ( ( (aob)~x)~b)o(((aob)oy)~b) } }o(aob). Proof: Apply Corollary 2.13 with p(x. y, z) = (xoy)o(zox) and note that the only binary term functions are x. y. xoy. Notice that the rquation with (ri. r2) = (b. b) is always vue and rhat for (h. a). (a&, a). (aoh. h) art: equivalent to these for (a. b). (a. aoh), (b, aob). respectively. Applying Theorem 5.2. Corollary 5.3 and Corollary 5.4 to Steiner quasigroups. GueIzow [2.4] ohtained the following theorem. Theorem 12.5. Let A = c A; O > be a tinitc Steiner quasigroup of nilpotence class k. Then there exists an m-dimansional vector space W. a polynomial p: Q2 22 Cl over GF(3). and a srqurnçe 1 6 ni < .. . < nt = m of integers such that (i) if 1 5 s < k and n, < i < n,+l. then pi(x. y) dors not depend on the variables Xn.+l. --• Xm "d y,,+1. .a- . Ym; (ii) pi(x, y) =O for al1 x. y é R and 1 l il nl; (iii) R = < Q: 0 > is isomorphic to A where xoy = -x -y + p(x. y); (iv) the center of A corresponds to the kemel of the projection ont0 the first nk.1 components of Q. This projection is a hornomorphism. From Theorem 12.5 we have the following theorem. Theorem 12.6. Let A = < A; O > be a finite Steiner quasigroup with order 3m. Then < A; O > is nilpotent if and only if it is isomorphic to < R: O >. where R is an rn-dimensional vector space over GF(3). xoy = -x -y + p(x. y), such that (il pdx. y) = O; (ii) pi(x. y) = pi(x l. .. . . xi. i, Yi, .. . . Yi. i) for 2 $ i m; (üi) p(x. X) = O. p(x. y) = p(yi X) and p(x. y) = p(xoy. x) for al1 x, Y E Rb Theorem L2.6 is quivalent to the following Grneralized Tripling Construction: Theorem 12.7. (Generalized Tnpling Construction). Let (A; be a nilpotent Steiner quasigroup of class n with order 3m. Let Bo, B,. BZ he a partition of the set of al1 3-element suhalgehns of A. R = A x GF(3). Define (x. i)dy. j) = ( XOY. i + j + h(x. y) 1. h(x.y) =O if x=yor Sg{x.y)~B~; A(x.y) =l ifSg(x.y}~B,; A(x,y) =2 if Sg{x.y}~B~. Then (R: 0) is a nilpotent Steiner quasigroup of clus n +l or n. Theorem 12.8. A finite Steiner quasigroup is nilpotent if and only if it is isomorphic to a Steiner quasigroup ohtained from the 3element Steiner quasigroup by repeated application of Generaiized Tripling Construction. The following lemma can be easily proved from Theorem 12.7. Lemma 12.9. Let A = (GF(3))2. Define xoy = ( - xi - yi. - X2 - Y2 + ~20%YI) such that < A, 0 > is a Steiner quasigroup. Then p2(x I. y 1) = c (x1 - yl)2 where c E GF(3). and then < A, O > is medial. In facf it is well known and easy to show that there is up to isomorphism only one Steiner quasigroup of sizr 9. namely the affine plane over GF(3). Let A = (GF(3))-'. Define xoy = ( - xi - yi. - x2 - y~.- ~3 - y3 + p3(~1.~2:YLYZ) 1- In the following wc: $ive some sprcial potynomials p3(xl,x2: y1.y2) such that < A, O > is a Steiner quasigroup. Example 12.10. p3(xl.ic2: y1,y2) = x2 yz(l + x2 y2)(xl - yl)2 for which the distributive law does not hold. Proof: Since zo(xoy) # (zox)o(zoy) for z = (0,1. O). x = (1.0. O). y = (1. 1.0). Example 12.1 1. pl(xi.xz: y i.yz) = xz y2(xZ+ yZ)(x - for which the distributive law does not hold. Proof. Since z~(x~y)+ (z~x)~(z~y) for z = (O. 1.0). x = (1.0.0). y = (1. 1.0). Example 12-12. (Guelzow [2]). pS(x1,x~:y1.y2) = xz y2(1 + x~ )(1+ yZ)(x1- ~1)~for which the distributive law does hold. Proofi Since zo(x0y) + (zox)~(zoy) for z = (O, 1,O). x = ( 1,O. O), y = (1. 1.0). Example 12.13. p3(xi.x2: yi.y2) = (xi - yL)(x2- yz) for which the distributive law holds. Example 12.14. p3(x1.~2: yI.y2) = (xl - yl)(xly2- xly ,) for which the distributive law holds. It is known thot the only distributive Steiner quasigroups of size 27 are also medial and so isomorphic to the 3-dimensional affine geometry over GF(3). ProoJ Since ZO(XOY)f; (ZOX)O(ZO~) for z = (O, 1.0). x = ( 1.0.0). y = (1. 1.0). Lemma 12.16. Let A be a 3-dimensional vector space over GF(3). Let xoy = - x - Y + p(x. Y) and pl(x. y) = p2(~.y) = 0- If p3(x. y) = [ a (~2- y212 + P X2y2( 1 + X2 yd + K X2 yAx2 f yd1 (x, - yd2. where a # O or P IL O or K F O. then A = c A; O > is a subdirectly irreducible nilpotent Steiner quasigroup of class 2. Prou$ From Example 12.10. Exampie 12.1 1 and Example 12.15. we have c A; O > is a nilpotent Steiner quasigroup. To prove ç A: O > is subdirectly imducihlr nilpotent of class 2. it is sufficient to show that c(A) = a= ( < x, y > : xi =yi. for 1 S i 12 ). Forthis we only need to show that the distributive Iaw holds iff a = P = K = 0. Wetake x = (1.0.0). y = (-1.1.0) and z = (O. 1.0). From (x0z)oy = (xoy)o(z~y), we have a + P + K = O; From (yoz)~x= (yox)~(zox), we have a + P = O. We take x = ( 1. O. O). y = (1, 1.0) and z = (0.- 1. O). From (y0z)ox = (yox)o(z~x), we have a = O. Hrnce a = B = K = O. Theorem 12.17. Let A be an (m+l)-dimensionai vector space over GF(3) with xoy = - x - y + p(x, y). Let a r O or B + O or K f O and definr Wl. y11 = a Ml- yd2 + p Xl ydl+ Xl yd + K XI y@, + y,): pl@. y) = pAx. y) = 0; pdx, y) = r(x1. YI)(XH - yt-~)~[l-pk-~(x. for 35klm+l. Then < A; O > is a subdireclly irreducible nilpotent Steiner quasigroup of class m. Then we have. for al1 x. y E A, (i) p(x. x) = O; (ii) p(x. y) = p(y, XI; (iii) p(x, y) = p(x0y. x); Thus, < A; > is a nilpotent Steiner quasigroup. TO prove < A: a > is subdirectly irreducible nilpotent of class m. it is sufficient to show that <(A) = { < x, y > : xi = yi, for 1 c i 5 m ). For this we only need to show that < O, b > E r(A) iff bi = O for 1 S i lm. By induction, we may assume that bi = 0 for 1 SiSm - 1. Let b = (0. ... .O. b,, t~,,~) and < O. b > E c(A). Frorn (iii) of Lemma 12.4 we have xoy = [ ( 0ox)ob Io[( (Oob)oy)ob) for ail x. y e A, that is. (*) (xoy)o{ (0ox)oh } = ( (Ooh)oy)ob for dl x, y E A Taking x = O . y = ( 1. O. . . .. 0) we have (~oy)o{(0ox)oh } = (Ooy)c{ (0oO)ob } = (Ooy)o( Oob } = (-y + p(O. y)}o{-b+p(O. b)} = (-Y )O{-b } = y + h + p(-y. -b) [( (00b)oy)obj = (-h + p(0, b) )oy)ob = (-h)oy)ob = (h -y + p(-b. Y) lob = (-h +y- p(-b. y) -h +p( b -y +p(-b. y). b) = Y + h - p(-b. y) + pt b -y. h) = y + b - p(-b. y) If a = O, taking x = y we have = -b+y -b+p( b-y, 6) = y +b sincr a = 0, r(-y,. 0)= 0 = X +b From the equation (+). we have that is, r(xI. x I) bm2 = O that is [ xi2 (1 + xi2) - K X, ] bm2 = O. By taking xl= 1 or-1. wr have Pbm' = K bm2 = 0; since a # O or p # O or K + O, we have b, = 0. On the other hand it is readily checked that Ç O. h > E c(A) for b = (0, . . . .O, 1) by Lemma 12.4. Thexfore we hava compketed the proof. Theorem 12.18. Let A be an m-dimensional vector space owr GF(3) and R = A x GF(3). Let (A: O) be :a suhdirecdy imducible Steiner quasigroup of class n with xoy = - x - y + p(x, y), w here (iii) p(x, y) = 0 if xi = O for 2 1 i Im. Then (R; 0) is a suhdirectly imducihle nilpotent Steiner quasigroup of class n+l. Prou$ The proof is similar CO that of Theorem 12.17. 13. Steiner loops Definition 13.1. An ulgebruic hop is an algehraic quasigroup with identity. A combinatoriul loop is a corn hinatonal quasigroup with identity. Definition 13.2. A Steiner loop is an algebra that has a binary operation " Q " with identity O satisfying the following identities: (i) x@x = O: (ii) XW = Oû3x = x; (iii) x@y = y@x: (iv) x@(x@y) = y. Lemrna 13.3. A Steiner loop is a combinatorid loop. Lemma 13.4. (Quackrnbush [19]). Let < A, 8,O > be a Steiner loop. Then a c(A) O if and only if for al1 x. y E A, (a. x. y} generates an associative subloop of A. Proof. Apply Corollary 2.1.10 with p(x. y, z) = (x@y)bz and the only binary term functions are x, y. xû3y. Lemma 13.5. Let < A. 8.0 > be a Steiner loop. The hllowing are equivdent : (i) < A, û3, O > is abelian; (ii) < A, @. O > satisfies the medial law; (iii) < A, €B. O > satisfirs (x@y)$(x@z) = y@z; (iv) < A, Bf. O > satisiïrs xQ(yû3z) = (x@y)Qz. Similar to tinite nilpotent Steiner quasigroups. wr have Theorem 13.6. kt A = < A, CD, O > k a tïnite Steiner loop with order 2m. Then < A, $. O > is nilpotent if and only if it is isomorphic to < R, o. O >, where R is an m-dimensional vector space over GF(2). xoy = x + y + p(x. y). such that (il pdx. y) = O; (ii) pi(x*y) = pi(x,, .. . , xi-1. y!. .. .. Yi-,) for 2 2 i 5 m; (iii) p(x. O) = p(x, x) = O. p(x. y) = p(y, x) and p(x. y) = p(x~y,x) for al1 x. y E Q. The following ûeneraiized Doubling Construction is classical. Theorem 13.7. (Generalized Doubling Construction). kt (A; B). O) be a Steiner ioop and B a set of delement subalgebns of A . Let h be the characteristic function of B. Q = A x GF(2), and (x. i)o(y, j) = ( xey. i + j + h(Sg{x. y}) ). Then (fi; O. O) is a sloop. Moreover. if (A: @,O) is nilpotent of class n. then (a:0. O) is nilpotent of class n or n +l; and ker (ir) is a congruence on R and contained in its center. where x is the projection of R onto A. From Theorem 13.6 and Theorem 13.7, we have Theorem 13.8. A tinitt: Steiner loop is nilpotent if and oniy if it is isomorphic to a Steiner loop ohtained from the Zelrmant Steiner loop by repeated application of the Genenlized Doubling Construction. Lemma 13.9. Let A k a 2-dimensional vector space over GF(2). Let x@y = x + y + p(x. y) and pi(x. y) = O. If < A, 8.0 > is a Steiner loop, then pz(x. y) = 0; that is. < A, 8.0 > is abeliui. Lemma 13.10. Let A be a 3-dimensional vector space over GF(2). Let x@y = x + y + p(x. y) and pi (x, y) = pz(x. y) = 0. If < A, CD, O > is a Steiner [oop. Then p3(x. y) = O. or p3(x. y) = x~y2+ ~2~1;that is. < A, @, O > is abelian. It is well known that al1 Steiner loops of size 5 8 are abelian. Theorem 13.1 1. Let A be an (m+2)-dimansional vector space over GF(2) with Proof: See the pmof of Theorem 16.4. Note the construction in Thereom 13.1 1 can be viewed as a specid case of the Generaiized Douhling Construction. To see this. let A,, be a 2-dimensional vector space over GF(2) with xoOy = x + y , and Chapter 5. Co-ordinatizations of Steiner Quadruple Systems Annanious and Guelzow [ 1.31 obtained structure theorems for finite nilpotent Steiner skeins. Guelzow (31 gave a construction of a Steiner skein of nilpotence class n with all denved Steiner loops of nilpotence class 1. Armanious [ 11 gave a construction for Steiner skeins of nilpotence class n with al1 its derived Steiner loops of nilpotence class n. In this chapter we survey the main results on nilpotent Steiner skeins and give a new and simple construction. in polynomial fom. for Steiner skeins of nilpotence class n with al1 its derived Steiner loops of nilpownce class n. 14. Steiner skeins Definition 14.1. A Steiner quadruple systern is a pair (X,B), where X is a set of elements (called points) and B is a collection of subsets of six four (called blocks) such that each 3-subset of X appars in axactly ont: block of B. Definition 14.2. A Steiner skein is an algebra which has a temary operation p satisfying the following iden tities: (il p(x, x. y) = y; (ii) p(x. y. z) = p(x. 2. y). p(x. yi 2) = p(y. 2. XI: (iii) p(x. y, pky. z)) = 2. It is well known that there is a one-to-one comspondence ktween Steiner quadruple systems and Steiner skeins (see [20]). Definition 14.3. (Quackenbush). A Steiner skein is called semi-boolean if it satisfies the additionai equation: (iv) p(x. u. p(y. u. 2)) = p(p(x. u. y). u, z). Definition 14.4. A Steiner skein is called boolran if it satisIles the additional equation: (v) p(x. u. pof. u. 2)) = p(x. y. a. A Steiner skein is bookan iff it is nilpotent of class 1. and is semi-boolean. A Steiner skein is semi-hoolean iff every derived Steiner bop is boolem (sre [2.3]). Since the ternary operation p of Steiner skrins i~1Tis a Mal'cev trrm. the variety of dl Steiner skrins is ptmnutabb and then modular. From CoroUary 2.13 we have the following lemrna. Lemma 14.5. Let < A, p > be a Steiner skein. Then a c(A) h if and only if for dl x, y, z é A, the following equality holds : p(p(a. h. x). y, z) = p(a. b. p(x. Y. ZN. It is easy to see that the congmances of a finite Steiner skein arr: r~gularand uniform. that is. any congruence is uniquely determined by any of its congruence classes. and any two congruence classes of the sami: congruence have the same size. Hence by Lemma 6.5, we have Lemma 14.6. Let < A. p > bs a tlnite nilpotent Steiner skein of order. Then < A. p > is subdirectly ineducible if and only if (;(A) is a congruence with class size 2. 15. Representation of finite nilpotent Steiner skeins Applying Drfinition 5.1. Theorem 5.2, Corollary 5.3 and Comllary 5.4 to Steiner skeins, Guelzow [2.3] stated the hllowing theorem. Theorem 15.1. Let A = c A: p > be a finite Steiner skein of nilpotence class k with order 2m. Then there exists an m-dimensional vector space W. a polynomial p: R3+ R over GF(2). and a xquance 1 $ ni < .. . c nk = m of integers such that (i) if 1 5 s c k and n, c i S n,+l. then pi(x. y) does not depend on the variables X*+l. -.-9 Xrn ad yn*+i. -.-. ym; (ii) pi(x, y. z) = O for al1 x. y, z iR and 1 5 i ln 1; (iii) = < Q: r > is isomorphic to A where ~(x.y. 2) = x + y + z + p(x, y. z); (iv) the center of A corresponds to the kemel of the projection onto the first nk-1 componentî of R. This projection is a homomorphism. (v) A is subdirectly imduciblr if and only if nk-~=m - 1. From Theorem 15.1 we have Theorem 15.2. Let < A; p > be a finite Steiner skcin with order 2*. Then < A; p > is nilpotent if and only if it is isomorphic CO < R; T >, where R is m m-dimensional vector space over GF(2) and r(x, y, z) = x + y t z + p(x. y. z). such that 52 Theorem 15.3. (Generalized Doubling Construction). Let (A; p) be a Steiner skein and B aset of 4clrment subalgebrasof A. Let )c bethecharacteristic functionofB. = Ax GW),and Wx. il. (y, j). (z. k)) = ( p(x. y. z). i + j + k + h(Sg(x. y, z}) ). nien (Q: Z) is a Steiner skein. Moreover. if (A; p) is nilpotrnt of clw n. then (R; s) is nilpotent of class n or n +l; and ker (rr) is a congnience on R and contained in its center. where x is the projection of R onto A. Theorem 15.4. A finite Steiner skzin is niipotcnt if and only ilit is isomorphic to a Steiner skein obtained hmthe 2-element Steiner skrin hy repeard application of the generalized dou bling consmction. 16. Recursive construction for Steiner skeins of nilpotence class n with al1 derived Steiner loops of nilpotence class n The following is a construction of a Steiner skein of nilpotence class n with al1 denved Steiner loops of nilpotençe clas 1 given by Gualzow [3]. Theorem 16.1. Let (A; p) ht: a Steiner skain and B a set or al1 Calement subalgebras of A. kt h be the characteristic function of B. and R = A x GF(2), Define Nx.i). (y. j). (z. k)) = (p(x. y. 2). i +j+k + h(Sg(x. y. z}) )- Then (Cl; 7) is a Steiner skein. If (A; p) is semi-boolean then (Q; r) is semi-boolean. Moreover, if (R: r) is subdirectly irreducible and nilpotent of class n+l if and only if (A; p) nilpotent of class n. The following is an example of a Steiner skein of nilpotence class 2 with al1 its derived Steiner loops of nilpotrnce class 2 given by Armmious [ 11. Lemma 16.2. ktA be a 6dimensional vector space over GF(2). Define p(x. y. z) = X + y + z + p(x, y. 2); r(x. y, z) = x1y2 + yp2 + ZlX2 + XlZ2 + YlX2 + z1y2; p1(x. y. 2) = p2(x, y, z) = p3(x, y. z) = 0; p4@. y. 2) = 0. y, z)(x~+ y3 + Z3 + x3y3 + y323 f ~3~3). Then < A; p > is a suhdi~ctlyirreducihle nilpotent Steiner skein of class 2 with all ifs drrived Steiner loops of nilpotence clas 2. Theorem 16.3. (Armanious [II). kt (A; p) bt: a finite suhdimctly irreducible nilpotent Steiner skein of clÿss n with dl its derived Steiner loops of nilpotence class n. Let A = { zi : OSi ~2~-'-1}, R= AxGF(2), [zziJ c(A) = ( zzi. Zzi+ 1 } , B = { { Qiv Zqi+l,Z4i+~*~4i+~} : O 5 i d 2m-2-l }, h be the characteristic function of B. T((x. i), (y. j). (rk)) =(p(x. y.~). i +j +k+ h(Sg{x. y. 2)) 1- Then (Cl; T) is suhdirectly irreducible and nilpoient of class n+l with dl its denved Steiner loops of nilpotence class n+l. In the following we give a new and simple constxuction in the fom of polynomiais. Theorem 16.4. ktA be an (m+2)-dimensional vector space over GF(2) with Then < A; p > is a subdimtly imducible nilpotent Sainer skein of class rn with al1 its denved Steiner loops of ni~po~ncrclass m. Proof. Fist we show that < A; p > is a Steiner skain. It is easy to sea that r(x. x, y) = O; so we have p(x. x. y) = y. Clearly. p(x, y. z) = p(x. z. y) = p(y, z. x). TO show ~(x.y. p(x, y, 2)) = Z. ~t:only need to show pi(x. y, Z) = pi(x. y. p(x. y. z)) for 1 5 i Im+2. Clearly it is tme for i = 1, 2. Since pi(x, y, z) = x 1 + y 1 + zi and p2(x, y. Z) = xz + y2 + zz. we have pl(x. y. 2) = p3(x. y, p(x, y. z)); that is. it is true for i = 3. Assume that it is me for i = k ( 2 31, then To prove A = < A; p > is nilpotent of class m. it is sufficient to show that c(A) = { Let b = (bt. b2, ... . b,+2) and O c(A) b. Frorn Lemma 14.5. we have (*) p(p(O. b.x).y.z) = p(0. b. p(x. y.z)) forail x. y. z E A. Assume b = (0. . . . ,O. b,+i. b,+2) and O c(A) b. We have = b + x+ p(0, b. x) + y + z + p(p(0. b. x). y. z) = b + x + y + z + p(h+x. y. z) (since hi = h2 = O. r(0. b. y) = O); From the equation (*), we have and then, b,+l = 0. Finaily. we show that al1 its denvsd Steiner loops are of nilpotence class m. Let c E A, we define xû3y = p(x, y. c) for al1 x. y E A. and we show that < A: @, c > is nilpotent of class m. It is sufficient to show that c(A) ={ For this we only need to show that if b c(A) O then bi = O for 1 L i S m+l. Let b = (hi. h2. -... bmcz) and c c(A) b. From Lemma 13.4. we have (x@y)bb = xe(y8b) for dl x. y é A. that is, (**) p(p(x. y. c). b. c) = p(x. c. pfy. b. c)) for al1 x. y e A. Taking xi = y 1. x2 = y1 and x3 = y3 = I+ c3. we have r(y. b. c) = O since p4(p(x. y. c). h. c) = p4(x. c. p(y. b. ç)). and ihen hi = c 1 and b2 = c2. p(x. c. pw. h. c)) = p(y. b. c) + x + c + p(p(y. b. c). x, c) = x + y + c + p(y, b, c) + b + c + p(p(y. b. c). x. c) = x + y + b + p(p(y, b. cl. x, c) since (bt, b2) = (cl, cd, and then r(p(x, y, c), b, c) = 0; this forces bm+i = cm+[- Theorem 16.5. Let A br an (m+l)-dimensional vcctor space over GF(2) and S2 = A x GF(2). Let (A; p) k a nilpotrnt Steiner skein with p(x, y. z) = X + y + z + p(x. y* 2) "d (i) pl (x, y, z) = O and (ii) pi(x, y. Z) = pi(x i, .. . . Xi. , Yi, .. . . Yi-1. zl, .. . . Zi. 1) for 2 B i 5 m+ 1. Then < $2: r > is a nilpotent Steiner skein. where T((x. i). (y. j). (2. k)) = ( P(X. y. z). i + j + k + P~+z(x*Y. Z) ) and (iii) pm+2(xV Y. 2) = Pm+ 1 (x. Y* Z)(XUI+ IYm+ 1 + Ym+lZm+ i + ~m+i~m+ i 1- Furthemore. if (A: p) is suhdiectly imducible Steiner skein of class n with al1 its derived Steiner loops subdirectly imducible and of nilpotence class n. then (R; T) is subdirectly imducihle and nilpotent of class n+I with al1 its drnved Steiner loops subdirectly imducihlr and of nilpotence clms n+ 1. Proof. The proof is similar to that of Theorem 16.4. Chapter 6. p-Groups General repwntation theorem for finite nilpotent groups and some examples of representations of finite p-groups for p = 2 with small order have been given by Gudzow in 121. In this chapter. we will mpresnt the dihednl group Dr and the generalized quaternion group Qzn which have maximal class. by recursive construction. 17. Definitions and basic results The foUowing definitions and results cmhe found in (211. Definition 17.1. If p is a prime. then a p-group is a group in which every element has order a power of p. Lemma 17.2. A finite group is ap-group if and only if the order of G is a power of p. Lemma 17.3. If G + {eJ is a finite p-group. then its center Z(G) * {e) . Lemma 17.4. If p is a prime, then rvery group G of ordrr is abdian. Lemma 17.5. If p is a prime and G is a non abrlian group G of order p3. then I Z(G) I = p. G/Z(G) G qx&- Theorem 17.6. Evrry finite p-group is nilpotent. Definition 17.7. A group G is said to ba a torsion group if every g E G has a finite order. Definition 17.8. The quaternions is a group Q of ordrr z3 genented by elernents a and b such that ab = e. b2 = a2 and ba = a3b. Definition 17.9. If n 2 3. a generalized quaternion group QZn is a group of order 2" generated by elernents a and h such that a2"-' = a. h 2,- a 2n-2 and ba = db. Definition 17.10. Lf n 2 1. a dihedral group D2n is a group of order 2" generated by elemenu a and b such that a2*-' = e. b2 = e and ha = db. Lemma 17-11. QzdZ(Q2n) a D2n-1. Lemma 17.12. Let G bt: a non empty finite set with an associative binary operation such that for al1 a. h. c e G. ab = ac a b = c and ha = ca b = c (cded the cancellation law). Then G is a group. Theorem 17.13. A group of order pn has a cyclic maximai suhgroup if and only if it is of one of the I'ollowing type : a cyclic group of order pn; the direct product of a cyclic group of order $'' and one of order p; the dihedral group D2n, n 2 3; the generalized quaternion group Q2n. n 2 3; 2 1 ,n-2- ' the sernidihedral group < x. a l x = L: = a . xax-' = a- >. n 2 3. Let G be a fmitely generated torsion group. If G is abelian, it is easy to show that G must b<: finite. In 1902. Bumside raised the following provocative question: Geneml Burmide Problem :Let G be a finitely generated torsion group. 1s G necessaily finite? Restricted Bumside Problem :Let G be a finitely generated torsion group of bounded exponent. 1s G necessarily finite? The answer to the General Bumide Problem is "no" in general. In 1964, Golod showed that. for any prime p. there exists an infinite p-group G generated by two elements. As for the Restricted Burnside Problem, the full mswer is not completely known. If wr let n ht: the exponent of G, the answer to the Restricted Bumside Problem tums out to depend on n. For n = 2. the answer is clearly " yes " as G must be abelian. For n = 3.4.6. the answers are still " yes ". by ~sultsof Burnside (1902). Sanov (1940). and M. Hall ( 1950). For n odd and 2 438 1. the negative answer to the Restricad Bumside Pmblem appeared in the work of Novikov and Adjan in 1968. Subsequent work of Adjan showed that thex is a negative answer to it for n odd and 2 655. Very recently. Ivanov (see [22]) has proved that there is a negative answer to it for al1 sufficiently large exponents n whathar even or odd. For srnail values of n. apparently not much is known. In particular, the case for n = 5 is sûll open. Theorem 17.14. Let G be a finitely generated torsion group. If G is nilpotent then G is finite (see [22]). Corollary 17.15. No finitely genented infinite torsion group is nilpotent. 18. Finite nilpotent groups with maximal class Theorern 18.1. The dihedral group D2n+i is a nilpotent group of class n for n 2 1. Proof: This is a standard exercise in group theory. Let m = 2"- 1. Letaandz genente D, and satisfy the following relation: (i) a2 = e. (ii) th = e. (iii) ar = +'a. First it is tnie for n = 1. For 1152. welet o = a'$ E Z(G).where O 5 i Il. O 5 j 12m - 1; - 1 then cm = tu. and then ai& = rai& and then air = rai. Since aT =r a and r t r-' we have i =O. That is. o = 4. Frorn wa = au, we have $a = a& and then t'a = r-'a. so that r2j = e. Hence Z(G) = (e. s *). and thrn G/Z(G) is a group of order 2x11 grmnted by [a]and [r] satisfr the following relation : (i) [a] = [el. (ii) [r] = [el. (iii) [a][r] = [TI -'[a]- Therefore by mathematical induction, we have compbted the proof. Corollary 18.2. For n 2 2. the genrnlized quaternion group Q. is a nilpotent group of class n- 1. Lemma 18.3. For rach n 1 2, let Gn be a finite p-gmup of class n. Dsfine H to be the group of al1 srquences ( gi. g2. . . - ), with g, e G, for al1 n and with g, = 1 for al1 large n; that is. g, z 1 for only a finia numbrr of g, . Then H is an infinite p-group which is noi nilpotent. 62 19. Recursive representation for D,o and Qtn Lemma 19.1. Let A = (GF(p))*. Define xoy = ( Xl+Yl. xn+ y, + p&~. x2. .-.. x,-1; yi. Y2. --- . hi)1. Then this hinary operation satisfies the cancellation law. xoy = (xi+ YI. x2+y2 + PZ(XI.Y~)1; X-L = ( XI. x2 + qdxd 1; e = (O. O). such that it is a group. Then pz(x1. y l) = cxlyl, q2(xi) = cxl. where c = O, 1. It is easy to sethat whcn c = O the gmup is Z2 x Z2 and when c = 1 it is 24. as listed in table 1. Table 1 Lemma 19.3. Let A = (GF(~))~.Define xoy = ( XI+ Yi. x2+Y2. X3f y3 + ~3(xl.X2:Yl.~2) 1; x-l = ( XI. x2. x3 + qj(x1. XZ) ); e = (O. O. 0). suchthat it is a group. Then p3(x.y) = axiyl + P x2y2 + hxly~+ ~XZYI, q3(x) = a xl + x2 + (h+ p)x1xZ. where a,p, AT p E {O. 1) (see the table 2). Table 2 (i) (a',p'. kt.p') = (a. a+ p+ A+ p. a+ h. a+ p) or (ii) (a'. b'. A'. p') = (a.p. A+ 1. p+ 1) or (iii) (a'. p'. k'. p') = (p. a. p. h) or (iv) finite composition of (i). (ii) and (iii). Theorem 19.4. Let A = (GF(2))". Let pi(x. y) = O. pz(x, y) = xlyi. PI& Y) = xk-iyk-1 + (~k-~+ yk- 1) P~-~(X.y) for 2 5 k In. qdx) = 0. qz(x) = xl. qk(x) = xk-1 [ XL-~+qk-I(~) 1 + qk-~(x) for 2 4 k In. Define xo y = x + y + p(x. y). e = (0. . . . .O). Then (A; O, -1, e ) is a cyclic group of order 2" with x-L = x + q(x) and u = (1.0. . . . .O) a gmerator. Prwf: It is easily checked that pk(x. y) for 1 S k < n a~ the cany functions for addition modulo 2" based on addition rnodulo 2. Lemma 19.6. Let A = (GF(~))~.Define xoy = ( xi+ Yi. x2 + yr x3 + Ys + X1Y2 + XzY2 1- Then (A; is a dihedral group such that for r = ( O. 1.0) and a = ( 1.0.0). Lemma 19.7. kt A = (GF(~))~.Define xoy = ( XI+ Yl. xr+y2. x3+ y3 + XlY2 + XzY2, X4+Y4 +~3~3+~2~2(~3+~3)+~l~2(~2+ X3 + Y~+Y~)+xIY~1- Then (A; is a dihedral group such that for 7 = ( 0, 1.0.0) and a = ( 1. O, O, O). (i) r8 = a2 = ( 0.0, 0.0) and aor = ~7~a;(ii) rios = d + a for 0 5 i 4 7. Lemma 19.8. Let A = (GF(2))". Suppose xo y = x + y + P(x, y) define a group operation such that for e = (0,. . . .O), t = (0. 1.0,. . . .O) and a = (1. 0.0,. . . ,O). (i) t~~-'=d=eand aos=rloa: Proof: It is easy to xethat (a) H(x, y) = H(y, x); (b) H(+, ri) = H(rioa. a) = H(ri, Au)= H(rioa. zioa); (c) = &,T~-J = a + T~-J + H(d, ~i-J); and then we have (d) P.) = Hi.) by (iii); (e) P(T~.aoa) = H(+, 3)= H(ri. riou) from sio(rioa) = si+j+ a and (b); from (~i~a)~zi= si-j + a and (c),and (b); (g) P(zioa, &a) = H(xi-J,J) = H(+-j, dos) = H(&aodoa, dos) from (~i~a)~(Zj~a)= ~i-Jand (c), and (b). Therefore. wt: have P(x. y) = H(x. y) + { H(x. y) + H(xoy. y) 1x1. Theorem 19.9. Let A = (GF(2))". Define xoy = x + y + P(x. y) where P(x. y) = Mx, y) + ( H(x, y) + Wxoy. y) 1x1. Then for e=(O,..., O), r=(O,1,0,..., O), a=(1,0.0..... O), (ii) ri0a = T~ + a; (iv) (A; ) is the dihedral group D2n; Proof: We only show that the associative law holds hy mathematical induction. For this. we show by induction that (i) P~(x*Y) + pk(x0y~z) = pk(y9 2) + P~X,Y~z); First. sincr pl(x. y) = ps(x. y) = hl(x. y) = h2(x, y) = 0. we have (i) and (ii) for k = 1, 2. Second, since h3(x, y) = x2y2, md hs(xoy, y) = (x2 + ~21~2.and then Thus, we have (i) and (ii) for k = 3. Finally, we assume that (i) and (ii) hold for k = m and we show that (i) and (ii) hold t'or k = rn +i. 68 Y) { pk(y. z)yl + pk(xoy. z)yl + pk(xoy. z) pk(x. y) + pk(x, yoz) fi(y. 2) 1 = O, where a(x- Y) = ( Yi +xUk +xlyk+ylxk +ylyk +yixl +xk zk +yk zk + xlzk 1. b(x, y) = ( xkzkyk + XlZkyk + XkZkyl+ YkZkYl + ZkYl + XlZkyl 1, and C(X,Y)= { (X~YL+~~l+~l~l+~l +~k~l+~l~k~k+~l~~~k+~k~~k+~l~k~k~- We have completed the proof. Theorem 19.10. Let A = (GF(2))", n 2 3 and L(x. y) = (O. . . . . O, xly ). Then for t=(O,1,0,.... O), a=(l*O-O...., O), n- 1 (i) 8=e, 52 =a2 and aor=r-los; (ii) &a = ri + a; (iv) (A; O ) is the generalized quaternion group Q2". My Publications [ 13 Zhang Xuebin, On the existence of (v.4.I)-PMD, Ars Cornbinatoria 29 ( 1990). 3- 12. [2] F. E. Bennett., Zhang Xuebin and Zhu Lie. Pe#ect Mendelsohn designs with block size four. Ars Combinatoria 29 (1990). 65-72. 131 F. E. Bennett and Zhang Xuebin, Resolvable Mendelsohn designs with block size four. Aequationes Mathematicae 40 ( 1990). 248-260. [4] Zhu Lie. Du Beiliang and Zhang Xuebin. A few more RBIBDs with k=5 and A = 1. Discrete Mathematics 97 (199 1). 409-4 17. [5] Zhang Xuebin. Constructions of resolvable Mendelsohn designs. Ars Combinatoria 34 (1992). [6] Zhang Xuebin, Consmctions for perfect threshold schemes, Corn binatorics and Graph Theory. (Hefei. 1992). 87-90. World Science Publishing. River Edge, NJ. 1993. 171 Zhang Xuebin, Indecompsuble triple system with A =.S. Journal of Combinatorial Mathematics and Combinatorial Computing 16 (1994). 153- 162. [8] Zhang Xuebin, Construction for in&composable simple ( v.4, A)-BIBDs. Discrete Mathematics lS6 (1996). 3 17-322. 191 Zhang Xuebin. On the existence of (v.4.l )-RPMD, Ars Combinatorh 42 ( l996). 3-3 1. [LOI Zhang Xuebin. Comtruction of orthogonal group divisible designs. Journal of Combinatorid Mathematics and Cornbinatonal Computing 20 (1996). 12 1- 128. [Il] Zhang Xuebin. Direct comtnictiun metho& for incomplrte pe rf ct Mrndelsohn designs with block size four, Journal of Corn binatond Designs. 4( 1996). 1 17- 134. 1121 F. E. Bennett and Zhmg Xuebin, Holey Perfect Mendelsohn dosigns of type mh with bbck size four, Journal of Combinatorid Designs. 5 ( 1997). 203-2 13. [13] Kong Gaohua and Zhang Xuebin. On the existence of (v,n, 4. AI-IPMD for even A, Ars Combinatorin. to appear. Note that my work of the last th= papes above was done at Department of Mathematics and Asuonomy of University of Manitoba. Bi bliography 1. M. H. Amanious. Nilpotent SQS-Skeins with nilpotent derived sloops. preprin t. 2. A, J. Guelzow, Some ches of E-Miniml Algebras of Annu Type: Nilporent squags. p-group and Nilpotent SQS-Skrins. PkD. Thrsis, University of Manitoba, 199 1. 3. A. J. Guelzow, The strucmre of Nilpotenr Steiner Quadruple systems . lournd of Combinatond Designs. 1 ( 1993). 30 1-32 1. 4. A. J. Guelzow, Representation offinite nilpotent squugs , Discrete Math. 154 ( 1996), 63-76. 5. J. H. Dinitz and D. R. Stinson, Contemporrrry Design Theory : A Collection of Survevs . New York. NY : Wiley, 1992. 6. C. J. Colbourn and D. R. Stinson, The CRC Handbook of Combinaturial Designs. Boca Raton. F'L : CRC Press. 1996. 7. R, S. Freese and R. N, McKenzie, Comtator theory for congruence modulor varieries. Cambridge University Press. 1987. 8. G. Gratzer, Universal Algebru. New York : Springer-Vdag. 1979. 9. H. P. Gumm, An easy wav to the comrnutator in duiarvarieties. Arch. Math. 34 (1980). 220-228. 10. H. P. Gumm, Geometrïcal muthods in congruence modulur algebrus. Mernoirs of die Amencan Math. Society, Vo1.45. no. 286, (1983). 1 1. B. Ganter and H. Werner, Equationai classes of Steiner svstems, Algebra Universalis 5 (1975). 125- 140. 12. C. Herrmann, Aflne algebras in congruence modular vurieries, Acta Sci. Math. 41(1979). 119- 125. 13. J. Hagemann and C. Hrmann, A concrete ideul multiplication for algebraic system and its relation to congruence distributivity. Arc h. Math. 32 ( 1979). 234-245. 14. C. C. Lindner and A. Rosa, Steiner quadruple system: a survey. Discre te Mathematics 2 1 ( 1979). 147- 18 1. 75 15. S. Klossek, Kornmutative spiegelungsrüume, Mitt. Math. Sem. Gia(kn 1 17( 1975). 16. R. N. McKrnzie. G. F. McNulty and W. F. Taylor, Algebras, Luttices, Varieties. Wadsworth & Brooks. 1987. 17 N. S. Mendelsohn, A namrczf generalization of Steiner triple system. in Cornputers in Nurnber Theory. Academic Press, New York, 197 1.323-338. 18. R. W. Quackenbush, Varieries of Steiner loups and Steiner quosigroups. Cmada i. Math. 28 ( L976). 1 187- 1198. 19. R. W. Quackenbush, Algebruic sprculutions ubout Steiner system, Annals of Discrete Mathematics 7 ( 1980). 25-36. 20. R. W. Quackenbush, Niipotent biock designs I: Basic concepts for Steiner triple und quadruple systems, preprin t 2 1. J. J, Rotman, An introduction to the theory of groups, S pnnger-Verlag. 1995. 22. D. J. S. Robinson, A course in the theory of groups. Springr-Verlag, 1996. 23. S. Sidki, On a 2-generated infinite 3-group, Journal of Algebn 110 (1987). 13-25. 24. J. D. H. Smith, Mal'cev varienes. kcture Notes in Mathematics. Vol. 554, Spnnger-Verlag. 25. W- Taylor, Som applicufiom of the term condition, Algebra Universalis 14 ( 1982). 1 1-24. 26. M. R. Vaughan-Lee. Nilpotentce in prrmtoble vurieties. kcturt: Noies in Mathematics. Vol. 1004, Springer-Verlag . 27. R. Mathon and Rosa, A census of Mt.ndel.sohn triple systemc of order nine, Ars Combinatioria. 4 (1977). 309-3 15. 28. F. E. Bennett, MendeLrohn triple sptemî without repeated bloch. Congr. Number. 20 (1977). 383-398. IMAGE NALUATION TEST TARGET (QA-3) APPLIED IMAGE. lnc 1653 East Main Street ,=-G ,=-G Rochester, NY 14609 USA --= --= -- Phone: 7161482-0300 =-= Fx71W208-5989