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Dissertations Graduate College

8-1974

The Automorphism Group of the Wreath Product of Finite Groups

Kenneth G. Hummel Western Michigan University

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Recommended Citation Hummel, Kenneth G., "The Automorphism Group of the Wreath Product of Finite Groups" (1974). Dissertations. 2917. https://scholarworks.wmich.edu/dissertations/2917

This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. THE AUTOMORPHISM GROUP OF THE WREATH PRODUCT OF FINITE GROUPS

by

Kenneth G. Hummel

A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of the Degree of Doctor of Philosophy

Western Michigan University Kalamazoo, Michigan August 1974

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS

Since it is impossible to list everyone who has helped, one way or another, along the road to completion

of this dissertation, I choose to mention here, since the

choice is mine, only the following people.

I thank Joe Buckley, under whose able guidance and

direction this thesis was developed; my wife Darla and my

children, Lacinda and Melinda, who bore graciously both

the good moods and bad which I alternately thrust upon

them during the period of my work at Western Michigan Uni­

versity; and lastly, the mathematics faculty and depart­ ment for their support, intellectual, moral, and financial.

Kenneth G. Hummel

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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. | 74-26,836 | HUMMEL, Kenneth 6., 1943- f; THE AUTOMORPHISM GROUP OF THE WREATH PRODUCT | OF FINITE GROUPS. | Western Michigan University, Ph.D., 1974 Mathematics

University Microfilms, A XEROX Company, Ann Arbor, Michigan

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS

CHAPTER PAGE

I PRELIMINARIES ...... 1

Introduction ...... 1 Definitions and Notation ...... 3

II THE AUTOMORPHISM GROUP OF THE WREATH PRODUCT ...... 6

Introduction ...... 6 Aut(A Wr B) as a Split Extension . . 7 The Subgroup H of Aut(A Wr B) . . . 16

III THE AUTOMORPHISM GROUP OF THEWREATH PRODUCT OF TWO P - G R O U P S ...... 41

Introduction ...... 41

The Subgroup H ...... 42 Some Results on the Structure of Aut (A wr B) ...... 53

IV THE CONJECTURE " |g | DIVIDES | A U T G | " ...... 58

Introduction ...... 58 G as a Central P r o d u c t ...... 58

G as a Wreath Product...... 64

i i i

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. THE AUTOMORPHISM GROUP OF THE WREATH PRODUCT OF FINITE GROUPS

Kenneth G. Hummel, Ph.D. Western Michigan University, 1974

An important concept in group theory is that of the

automorphism group of a group. It has been shown that the automorphism group of the standard wreath product W of

a group A by a group B is, except in a few cases, a

splitting extension of A*HI^ by B*, where A* and

B* are isomorphic to the automorphism groups of A and

B respectively, 1^ is a particular subgroup of the

group of inner automorphisms of W, and H, although

somewhat described, is really the "unknown ingredient" of

Aut W. In this thesis the subgroup H of Aut W is des­

cribed when the group A is purely non-abelian, abelian,

or cyclic, and when both A and B are p-groups. Come

of these descriptions yield formulas for the order of H, o(H), (hence for o(Aut W)) and methods whereby all the

automorphisms of W may be constructed. Furthermore, when both A and B are p-groups, a Sylow p-subgroup of

Aut W is described as a splitting extension. A similar

result is obtained for Out W.

The conjecture that o(G) divides o(Aut G) when G is a non-cyclic p-group of order greater than p 2 is investigated when G is a central product and also when

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G is a wreath product. Both investigations yield new

classes of groups for which the conjecture is true. Fur­ thermore, the central product work shows that the general

conjecture will be established if it can be proven for a

smaller class of groups.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I

PRELIMINARIES

Introduction

Most branches of mathematics introduce new and impor­

tant concepts into their fields of study when they define

"products of" and "operations on" particular objects in

their realm. In analysis, for example, products of func­

tions, spaces, and measures are important concepts as

well as operations such as differentiation, integration,

and summation. In graph theory there is the cartesian

product of graphs and operations such as complement and

converse. This thesis is concerned primarily with the

standard wreath product of groups (which is again a group)

and the operation "Aut" which assigns to each group G

another group, denoted by Aut G, consisting of all the

automorphisms of G. Chapters II and III investigate the automorphism group

of the standard wreath product, the combination of these

two important group theoretic ideas. A description of

this automorphism group has been given by Houghton [9],

but his description contains a subgroup H about which

little was known. Let W be the standard wreath product

of a group A by a group B. Chapter II contains a more detailed description of the subgroup H of Aut W when

1

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A is purely non-abelain, abelian, or cyclic. Chapter III

is an investigation of Aut W when both A and B are

finite p-groups. In this case a formula for the order of

H, o(H), and hence for o(Aut W) is determined. Fur­

thermore, the proof is constructive and yields a method

whereby all the automorphisms in H and hence in Aut W

may be realized from certain homomorphisms.

Chapter IV investigates the conjecture that o(G)

divides o(Aut G) when G is a non-cyclic p-group of or- 2 der greater than p . The conjecture is shown to be true whenever G is the central product of non-trivial proper

subgroups A and K, where A is abelian and o(K) di­

vides o(Aut K) . This result extends the classes of

groups for which the conjecture is known to be true and

also shows that if the conjecture can be proven for a par­

ticular class of p-groups, then it is true for all p-groups

under consideration. Also in this chapter, results from

Chapters II and III are used to yield theorems giving con­

ditions on the p-groups A and B under which the con­ jecture is true for W = A wr B . Examples of these con­

ditions are: o (A) divides o(Aut A) and o (B) ;> o(Z(A)),

where Z(A) is the center of A . Furthermore, it is

shown that the conjecture is true for the standard wreath

product of any two non-trivial p-groups A and B if a

weaker statement is true for all non-abelian indecompos­

able p-groups G .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Definitions and Notation

If A and B are non-trivial groups, the wreath

product of A by B is defined as follows. Let F^

be the group of functions from B into A with multi­ plication of f,g € defined by fg(x) = f(x)g(x) for all x 6 B. F1 is the unrestricted direct product of iso­

morphic copies of A, there being one copy of A for

each element b in B. The subgroup F2 of F^ consis­

ting of all functions with finite support is the corres­

ponding restricted direct product. A function from B

into A is said to have finite support if its value is

trivial at all but a finite number of elements of B .

Both of these groups will be denoted by F and the dis­

tinction made when necessary. If f € F and b £ B, then

f*5 £ F is defined by f^(x) = f (xb 1) for all x € B .

The group of automorphisms of F defined by f -* f^ for all f € F is isomorphic to B and will be identified

with B . The wreath product of A by B is defined as the splitting extension of F by B; i.e., it is gener­ ated by B and F with the relations b -1fb = f^ for

all b 6 B and f € F. if f is unrestricted the wreath

product of A by B will be denoted W = A Wr B . The

notation A wr B will be used when F is restricted.

F will be referred to as the base group of W. The sub­

group of F consisting of all constant functions will be

3

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. called the diagonal subgroup and denoted by D.

For any group G, Z(G) will denote the center of G.

In the case W = A wr B, P will denote that subgroup of

F defined by P = {f € Z(F) : xgB tf(x)] = where 1 is the identity element of A. It is noted that P is

the direct product of o(B) - 1 isomorphic copies of

Z(A) .

A group G will be called indecomposable if it can­

not be expressed as a direct product of two non-trivial

subgroups. For any group G, Aut G will denote the

group of automorphisms of G and AC (G) will be used to denote that subgroup of Aut G consisting of all the cen­

tral automorphisms of G. A central automorphism of G

is an automorphism f of G that has the property;

g_1f(g) € Z(G) for all g € G. Let G be a group and g 6 G. i^ will denote the inner automorphism of G de­

fined by ig(h) = 9 ^ 9 f°r h € G and Inn G will denote that subgroup of Aut G consisting of all the inner

automorphisms of G. Out G will denote the group of outer automorphisms of G. If H ic a subgroup of G (denoted

by H £ G), then AutHG consists of those automorphisms

of G which leave H invariant. If H £ G, then H is

characteristic in G (H char G) if and only if AutHG =

Aut G. For any two groups G and K, Horn(G,K) is the set of all homomorphisms from G into K. If K s G and

f is any map defined on G, then f|K is the restriction

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5

of f to K. A group G is said to be purely non- abelian if it has no non-trivial abelian direct factors.

The fact that two groups G and K are isomorphic will

be denoted by G =- K. The Frattini subgroup of a group

G is the intersection of all maximal subgroups of G.

The notation ft(G) will be used for this group and the

usual convention that $ (G) = G if G has no maximal

subgroup is adopted. Other definitions and notations will appear in the

text of this thesis when they are needed.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER II

THE AUTOMORPHISM GROUP OF THE WREATH PRODUCT

Introduction

The wreath product of two groups is playing an in­

creasingly significant role in group theory. Like other

"products" of groups (e.g., direct, central, and semi-

direct) it is a method whereby a third group may be con­

structed from two other groups. The third group or child

may, as is the situation in real life, have either great

or little "resemblence" to its parents. In the case of

the so-called "standard wreath product" of groups, the child does "inherit" many important characteristics from

its parents. These traits have been exploited at various

times by Neumann [12], Houghton [9], Neumann and

Neumann [11], and others to ascertain important new

information about the standard wreath product and its corresponding automorphism group. Some of these results

have been collected at the beginning of this chapter so

that this thesis might be a more complete entity in and of

itself. In some cases the theorems are restated and proofs

which are more in the flavor of the rest of the thesis are

given.

It is in this chapter that some new descriptions of

the subgroup H of Aut(A wr B) are obtained. The most

6

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. striking new result is contained in Theorem 2.13. Working

from the hypotheses that A is a purely non-abelian fi­ nite group and that B is a finite group, a one-to-one

map of H onto a set R is obtained. Furthermore, the proof is constructive and thereby yields a method by which all the automorphisms of W in H may be constructed from elements of R. A formula for o(H), obtained from

a formula for the cardinality of R, is given as a

corollary.

Aut(A Wr B) as a Split Extension

The initial theorem stated in this section is due to

Neumann [12, p. 368]. The result is significant in its

own right, essential to later work in this chapter, and

rather startling. The proof is fairly long and complex.

It is omitted since the result and not the technique of proof is the important factor in this thesis.

Definition. If A is a dihedral group whose normal abel­ ian subgroup K of index two contains unique square roots of all its elements, then A is a special dihedral group.

Theorem 2.1. The base group in a (restricted or unre­ stricted) standard wreath product of a group A by a group B is a characteristic subgroup except when B has

order two and A is a special dihedral group. In this

case the base group F admits a subgroup of index two in

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the automorphism group of the wreath product. In other

words, AutpW = {<* g Aut W : a (F) = F) is a subgroup of

index two in Aut W.

The next two lemmas are also basic background for

later results and were first given by Peter Neumann's parents, B.H. Neumann and Hanna Neumann [11, Pp. 474-6].

The proofs are routine in nature and are omitted.

Lemma 2.2. If a € Aut A, then a * defined by

(bf)a = bf^ for all b £ B and f € F, where fa*(x) = (f(x))a for all x € B is an automorphism of

W = A Wr B. Furthermore, the group A* of all such auto­

morphisms is isomorphic to Aut A.

A similar result for the automorphisms of B is given

by

Lemma 2.3. If 3 e Aut B, then p* defined by (bf)^* = b^f^* for all b g B and f £ F, where

f^*(x) = f(x^ ) for all x ^ B is an automorphism of

W = A Wr B. Furthermore, the group B* of all such

automorphisms is isomorphic to Aut B.

It is also noted at this point that the groups A* and B* commute elementwise.

The next theorem is due to Houghton [9, p. 309], but

is restated somewhat to better fit its role in this thesis.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Before stating this theorem it is convenient to intro­

duce some notation and definitions which will be used re­

peatedly throughout this chapter and the next. Let W =

A Wr B. I1 will denote the subgroup of Aut W consisting of those inner automorphisms corresponding to conjugation by elements of the base group F. K will be used to denote

the subgroup of AutpW consisting of those automorphisms

which leave B elementwise fixed. The subgroups A* and B* of Aut W are as defined in Lemmas 2.2 and 2.3 re­

spectively. Finally, H is the subgroup of K consis­ ting of those automorphisms which leave the diagonal ele­

mentwise fixed.

Theorem 2.4. Let W = A Wr B. Then except when B is

of order two and A is a special dihedral group

(a) Aut W is a splitting extension of KI^ by B*, (b) K is a splitting extension of H by A* ,

and (c) the subgroups A*HI1# HI-jB*, HI1 and ^ are all normal in Aut W. In the exceptional case (a) through (c) hold with

Aut W replaced by AutpW a subgroup of index two.

Proof; By the result of Neumann which is stated as Theorem 2.1 in this thesis, AutpW = Aut W except when

B is of order two and A is a special dihedral group. Hence both cases of the theorem can be proven simultan­

eously by showing that (a) through (c) hold with

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. io

Aut W replaced by AutpW for any choice of the groups

A and B. (a) AutpW is a splitting extension of by B*.

It is first noted that an element w £ W can be expressed

uniquely in the form w = xg where x € B and g € F, w that F is a normal subgroup of W and — =“ B. If w a € AutpW, then a induces an automorphism u(a) of —

w defined by u(a) (wF) = a(w)F for all wF 6 — . If p € Aut then for each b € B there exists a unique

x € B such that p (bF) = xF. Hence the map r(p):B -» B

defined by r(p)(b) = x for all b £ B where p(bF) = xF

is an automorphism of B. Consider the map t = r » u

taking AutpW into Aut B. Note that if a 6 AutpW, then

t(a) (b) = x for all b € B where a (b) = xg in canon­ ical form. If s: Aut B -» AutpW is the homomorphism de­

fined in Lemma 2.3, then tos(fj) = t(p*) = p and AutpW

is a splitting extension of ker t by Aut B “ B*.

The proof of (a) will be complete if ker t = KI^. First observe that a £ AutpW is in ker t if and only

if a(b) = b (mod F) for all b g B . Since K s AutpW and elements of K leave B fixed elementwise, it is clear that KI.^ c ker t. To show the other inclusion let

a 6 ker t. For each b £ B there exists an element g^

in F such that a (b) = hg^. Define g € F bY g(b) = [gb (b)]"'L for all b g B. Note that for all

x,y € B, a (xy) = *ygxy and (xy) = xgxY9y = *yg£gy .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11

Hence = 9x9y for a11 x,y € B. A special case is

gx v = 9 for a11 x € B. It follows that x“ X [ (g-1)13^ ] (X) = [g(xb-1)]_1gb (x)g(x) = g... _1 (xb_1)gb (x) [gx (x)]-1 xb = g£ (xb"1)g^_1 (xb"1)gb (x) [gx (x)]"1

= gx (x)g^_x (x)gb (x) [gx (x)3_1

= gx (x) tgb (x)] _1gb (x) [gx (x)l _1 = 1

for all x £ B. Hence (ig 0 ot) (b) = ig (bgb ) = g ^ 9 ^ 9 = b(g”1)t>gbg = b for all b £ B, where ig is the inner

automorphism of W corresponding to conjugation by g.

Therefore ig o a is in K and a 6 I-jK. A routine

argument shows that 1^ is normal in AutpW. Hence

I^K = KIX and ker t c which completes the proof

of (a) .

(b) K is a splitting extension of H by A*. Let d € D and p € K, then p (d) = p (d*5) = p (b ’^db) =

b _1p (d)b = p (d)b for all b £ B. But g q F satisfies

gb = g for all b £ B if and only if g 6 D. Hence for

all p € K, p (D) = D. Consider the map v:K -+ Aut A

defined by v(p) = P|D for p £ K. This can be con­ sidered as a map into Aut A since D =- A and hence

Aut D “ Aut A. If s:Aut A -» K is the homomorphism

defined in Lemma 2.2 it is a simple matter to observe that v o s is the identity on Aut A. Hence K is a

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12

splitting extension of ker v by Aut A A*. But p g ker v if and only if p|D is the identity on D.

Hence ker v = H, those automorphisms of W in K

which leave the diagonal elementwise fixed. (c) A*HI-, HI-B*, HI, and I. are normal in Aut„W. ± 1 1 1 r These normalities are routine computations and are left to

the reader. #

The next theorem is also a result of Houghton and be­

gins to give some insight into the nature of the subgroup

H. Although the theorem is actually a statement about K

it is recalled that H s: K.

Theorem 2.5. The group K is isomorphic to the group

of those automorphisms of the base group which commute

with the inner automorphisms of W induced by elements

of B.

Proof; Let p 6 K, b £ B, and f € F. It is clear that p (fb) = p (b-1fb) = b-1p(f)b = p (f)b and hence p commutes

with ife for all p £ K, b £ B. Conversely, if

a € Aut F commutes with i^ for all b £ B, then o/ can

be extended to an automorphism y of w which fixes B elementwise. Define y;W -» W by y(bf) = b<*(f) for all

b e B and f g F. #

The previous work has been presented to serve the

reader as basic background information. The rest of this

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13

Chapter will be devoted mainly to new results. Before

proceeding with the new results it is convenient to make

the identification suggested by Theorem 2.5/ i.e., the

subgroup H of Aut W will be identified with the sub­

group of Aut F consisting of those automorphisms of F

which commute with the inner automorphisms of W induced

by elements of B and which also fix the diagonal element­ wise. This identification will be seen to be notationally

useful in some of the following work.

The next lemma will be quite useful in obtaining a

description of the subgroup H when A is purely non-

abelian. It is also of some interest in its own right as

it shows that the subgroup H always contains a subgroup

J(B) which is isomorphic to B.

Lemma 2.6. If b £ B, then j ^ F "♦ F, defined by j^(f) = f(k) for a ;j^ f g f, where f ^ ( x ) - f(b^x)

for all x £ B, is in H. Furthermore, J(B) = {j^zbgB)

is isomorphic to B.

Proof: If f and g are in F and b 5 B, then

(fg) (b) (x) = fg (b-1x) = f (b^x) g (b_1x) = f (b) (x) g(b) (x) for all x € B. Hence g Horn(F,F) for all b £ B .

To show that is one-to-one suppose that jb (f) = (g) for some f and g in F. It follows that f ^ (x) _

g ^ (x) for all x £ B and hence f (b ^x) = g(b ^x) for all x £ B. Therefore f = g and is one-to-one

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14

for all b in B. Finally to show that is an iso­

morphism of F it remains to show that is onto. But

if f is in F, then f ^b ^ is in F and

jfc(f^b = [f^b h = f. To finish the proof that

J(B) £ H it is necessary to show

(i) jb 0 L x = l x ° jb for a11 b 'X € B and (ii) j^|D is the identity on D for all b g B,

where ix is the inner automorphism of W corresponding

to conjugation by x £ B. Proof of (i): If f £ F and b,x € B, then jfcoix(f) =

(fx) (b) . But for all y € B, (fx) (b) (y) = f ^ b " 3*) =

f(b“1yx“1) = f ^ (yx-1) = ( f ^ ) X (y). Hence j^oi^tf) = (fX) ^ = ( f ^ ) x = ixoj^(f) and (i) is established.

Proof of (ii): If d € D and b £ B, then = <3^bl But d^b '(y) = d(b_1y) = d(y) for all y € B. Hence

jfc(d) = d (b) = d. It is left to show that J(B) ~ B. If

f d F and x,y € B, then f (xy)(b) = f((xy)-1b) =

f(y-1x-1b) = f (y)(x-1b) = [f(y)] (x)(b) for all b € B.

Hence jxy(f) = f (xy) = [f(y >](*> = Dx (f(y)) = V i y (f) for all f g F and x,y 6 B. In other words jx^ = jx°jy

for all x,y £ B and t:B -♦ J(B) defined by t(b) = jb for all b £ B is a homomorphism. It is immediate from

the .definition of t that it is onto. Next let x,y £ B

and suppose that jx = It follows that Dx (f) = 3y(f) for all f £ F and hence that f ^ (b) = f ^ (b) for all

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15

f in F and b in B . In other words f (x‘ _1b) = f(y*"1b) for all f g F, b g B. In particular f(e) =

f(y-1x) for all f g F. Since A is non-trivial this is

only possible if y ^x = e. Hence jx = implies that

x = y and t is one-to-one. #

Throughout the rest of this chapter A and B will

denote finite non-trivial groups. Hence the wreath pro­

duct of A by B is written W = A wr B. The following theorem emphasizes the importance of

being able to obtain a formula for the order of H.

Theorem 2.7. Let A and B be finite non-trivial

groups and W = A wr B. If

X = |W| - | H| *| Aut A| • | Aut B|/|A| * | B|

then the order of Aut W is given by

Proof: It is sufficient to show that the order of KI^B*

is X. Since KI^B* is a splitting extension of KI1 by

B* ~ Aut B it follows that |KI1B*| = |KIX|-|B*| = |KlJ •|Aut B| and attention is shifted to the group KI^.

If f g F, then i^ is in K n I-j_ if and only if if (b) = b for all b in B. Now if (b) = f_1bf = b

for all b g B if and only if f = f^ for all b g B.

But f = for all b g B if and only if f g D. Hence

it follows that K n ^ = (if : fgD) ~ D/Z(D) ~ A/Z(A).

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore | KI1( = | K| -1 I11/| K ij

= |K| -| Ii| -|ZS(A) |/| A| .

If f,g g F, then if and only if g ^f £ Z(W) = Z(D) ~ Z (A) . Hence | I ± \ = |f|/|Z(A)| = | A| I B l / | Z (A) | . Next it is noted that | K| = |h|*|A*| - |H|*|Aut A| since

K is a splitting extension of H by A* =- Aut A. Com­

bining the above results it is seen that | KI-jB*) =

| KI1| -| Aut B| = | K| -1 I11 -1 Z (A) | - | Aut B|/|A| = | H| * | Aut A| - | A| I B l *| Aut B 1 /1 A| = | W| * | H| -| Aut A| -| Aut B|/|A| -1 B| = X. #

The Subgroup H of Aut W

Since H has already been identified with a sub­ group of the automorphism group of F, and since F can

be viewed as the direct product of o(B) isomorphic

copies of the group A, it is both natural and notation-

ally useful to make the following further identification.

Let f £ F, B = {b1, bn ) and suppose f(b^) = ai

where a.^ € A for all i = 1,2,...,n. Identify f with n . n (a.) i . It is noted, for further clarity, that under i=l 1 this identification each element a in A has been iden­

tified with the function f in F which assumes the

value a at the identity element of B and is trivial

elsewhere. Hence the use of ab to denote that function

in F which assumes the value a £ A at b € B and is

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trivial elsewhere is compatible with the action of B on

F. If L

trivial everywhere except, perhaps, at b. Clearly I?3 ** L

for any b £ B. The following result is an essential ingredient in

the proof of a number of later theorems. It is given at

this time so that the argument will not have to be repeat­

ed each time the result is needed. The proof uses the

Remak-Krull-Schmidt Theorem [17, p. 83], one version of

which states

Remak-Krull-Schmidt Theorem. If a finite group G has

decompositions G = A^ x ''* * Am = B1 * ’ * * * Bn w^ere the A^ and B^ are indecomposable and non-trivial, then

m = n and there exists a permutation cr 0 Sn and a cen­

tral automorphism y of G such that y (Ai) = B ^ .

More precisely, the proof uses the following easy

Corollary. If a finite group G has a decomposition

G = A^ x ••• X An and 0 0 Aut G then there exists a permutation CT 0 Sn such that 0(A^) = A ^ and

0 (A•) £ A. Z(G). l lcr

Also, the identifications given above and the properties of an element in H are used extensively.

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Lemma 2.8. Let A and B be non-trivial finite groups.

Let A = (Ai;l x X Alltl]L) X ••• X (Arl x X where A^.j is non-trivial indecomposable for all i and

j and A ^ =- Aj^s if and only if i = k . Let e denote

the identity element of A and P = {f^Z (F) :^ B [ f (x) ] =e }.

If cp € H, then for each direct factor A^^ of A there

exists an element b € B such that cp(A^j) £ .

Moreover, for each a € A^_. there exists an f € P such

that cp(a) = ab f .

Proof: Recalling that A*?j = {ab : a € A ^ ) it is seen

that F = X X X1 A?. . Since cp € H it follows x€B i=l j=l 13 r m± r mi that cp (F) = X X X cp (A. .) = X X X cp (A. .) = F. x€B i=l j=l 13 x€B i=l j=l 13 Fix i and j , then it follows from the Corollary to the

Remak-Krull-Schmidt Theorem that cp(A^.) £ A^Z(F) for

cp (a) = c^f . But cp € H and hence cp(ax) = cp(a)x = cbXfX ^ Furthermore, cp| D is the identity on D and so

X"B'aXl - " x S b I ^ - ^ b ' ^ ' Evaluating at the identity element e of B it is seen

that a = cz , where z = € Z(A) . Hence c = a z ’”'L and cp (a) = c^f = a ^ z -1)^

g = (z~1)'bf € Z (F) . A review of what has been shown to

this point shows that the proof will be complete if the

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function g £ Z(F) which depends on a 6 A.^ and is given by cp(a) = abg can be shown to be more precisely in

P s Z (F) . Recalling that cp (aX) = cp(a)X for all x £ B

and that cp|D is the identity on D it is seen that

X^ 1^ 1 = = ^BEtP<^)l

■ * " b M 3 >X 1 - X ^ B [ a b X l •

But x^B [ax] = x^B [abx] and hence x^B [gX] = 1, the iden-

tity of F. If e is the identity of A and y 6 B, then

0 = x € B [gX(y)] = x € B [g(yx"1)] = x<=B[g(x)] * Hence 9 € P and the proof is complete. #

In trying to describe the subgroup H of Aut W a typical difficulty is encountered, i.e., to obtain more

and more detailed descriptions of H it is necessary to

place more and more restrictions on the groups A and B.

The trick, obviously, is to balance one against the

other. The initial theorems in this chapter placed no re­

strictions on A and B and provided some information

about H. More information about H has been acquired

by assuming the groups A and B to be finite. In the

next lemma and theorem it is further assumed that A is

non-abelian indecomposable. As expected, even more de­ tailed information about H is obtained.

Lemma 2.9. If A is a non-abelian indecomposable group

and B = (b1,b2, ... ,bn ) , then CT:Hom(A,P) -» AC (F) n H

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defined by [ff(t)](n [(a.)b i]) = n [(a.)bi]• n [t(a.)bi] i=l 1 i=l 1 i=l 1 n b _ for all t g Hom(A,P) and n f(a.) i] g F is a one-to- i=l 1 one map of Horn (A, P) onto AC (F) fl H.

Proof: If t g Horn (A, P) , then t*:F -» P defined by

t*( nt(a. )bi]) = n [ t (a. )bi] for all n [(a. )bi] € F i=l 1 i=l 1 i=l 1 is an extension of t to a homomorphism of F into

P < Z(F) which commutes with the inner automorphisms of W corresponding to conjugation by elements of B. Now A is non-abelian indecomposable and hence F is purely

non-abelian. It follows by a result of Adney and Yen [1,

p. 137] that cr (t) :F -> Fdefined by [rr(t)] (g) = gt* (g)

for all g € F is a central automorphism of F. It is clear that o^t:) commutes with i^ for all b 6 B and

hence to show that ^(t) is in fact in H it is neces­ sary to show that

d € D and d(b) = a for all b g B, then cr(t) (d) = n b • d n [t(a) 1] = d since t(a) g P implies that i=l n - _ II [t(a) i] is the identity of F. It has been estab- i=l thus far that cr is, indeed, a map of Horn (A, P) into

Aq (F) f| H. It is next shown that cr is one-to-one. If

t and u are in Horn (A, P) and rr(t) = cr(u) then cr(t) (a) = cr(u) (a) for all a g A. But then at (a) = au(a)

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for all a £ A and t = u. Hence or is one-to-one. To

show that cr is onto let cp £ Ac (p) H H; since cp € Ac (p) » i°r each a 6 A there exists a g £ Z(F)

such that cp(a) = ag and in fact, g € P since cpj D is

the identity on D. It follows that t:A -» P defined by

t(a) = a-'Lcp(a) is in Hom(A,P). Since an automorphism

of F in H is completely determined by its action on

A/ o(t) = cp and or is onto. This completes the proof.#

The subgroup J(B) of H described in Lemma 2.6

plays an important role in the description of H when A

is non-abelian indecomposable. This is seen in

Theorem 2.10. If A is a non-abelian indecomposable

finite group and B is a finite group, then H is a

splitting extension of Ac (p) fl H by J(B). Moreover,

if B is abelian, then H = (Ac (F) n H) x J(B).

Proof: If cp € H there exists a unique b € B such that

for each a 6 A, cp(a) = a^f for some f 6 P. It is

noted that the existence of a b € B is given by Lemma 2.8 and its uniqueness by the fact that A is non-

abelian. Consider the map t:H -> J(B) defined by

t (cp) = where b is the unique element of B associ­

ated with cp as determined above. It is noted that

g € P if and only if g 6 Z(F) and = ‘L* Hence

for all cp 6 H, cp(P) = P. Now let cp^^ and cp2 be in H

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and let x and y be the associated elements of B for

cp^ and cp2 respectively. Let a G A, cp1(a) = axg, and

(02 (a) = a^f where f and g are in P. It follows that

cp1°cp2 (a) = cp1(ayf) = ^ (a)Ycp1 (f) = a ^ g ^ ^ f ) = aXyh , where h = g ^ U ) 6 P. Hence t (cp1oCp2) = = jx° jy =

t (cp^) 01 (cp2) anc^ t is a homomorphism. Now =

a^) = for all a 6 A and hence Mjj-j) = ^b ' s: J (B) -» H is the inclusion map then t° s is the iden­

tity on J(B) and H is a splitting extension of ker t

by J(B) . Now cp € ker t if and only if t(cp) = and

hence ker t = Ac (F) n H. If B is abelian = i^ for all b G B and hence

J(B) Z (H). It follows that when B is abelian

H = (Ac (F) n H) X J(B). #

Since P £ Z(F), it follows that Hom(A,P) is a

group for any choice of the group A. Recalling that

P = [fGZ (F) :x^B t f (x) ] = e), where e is the identity of A it is clear that Horn(A,P) is isomorphic to the direct

product of o(B) - 1 copies of Horn(A,Z(A)). When com­ bined with Lemma 2.9 and Theorem 2.10 , this observation

yields the following immediate result.

Corollary 2.10.1. If A is a non-abelian indecomposable

group and o(B) = n, then o(H) = n-o(Horn(A,Z(A)))n_1 .

A slight digression from the investigation of H is

appropriate at this time. It deals with the wreath power

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of a group and hence begins with the following definition.

Definition. Let A be a finite group. The n— • wreath

power of A is defined recursively by ^A = A and

nA = (n-1A) wr A for all n s 2.

This digression is appropriate at this time since nA

is non-abelian indecomposable for n :> 2, where A is

any finite group. This result follows from Theorem 7.1

in Neumann [12, p. 356] which characterizes those stan­

dard wreath products which are indecomposable. Combining

this result with another result of Neumann (which will be

given in the proof of Theorem 2.11) and using Corollary

2.10.1 and Theorem 2.7 which deal with the orders of H

and Aut W respectively it is possible to obtain

Theorem 2.11. Let A be an arbitrary finite group and

let denote the subgroup H of Aut (nA) for all n 2: 2. Then if | A| = m:

(a) |Hj = m-1 Hom(A/A' ,Z (A) ) | (n-1) (m_1) for n ;> 3, where A' denotes the derived (commutator) subgroup of A

and (b) | Aut (nA) | = j nA| - n | HjJ • | Aut A| n/| A| n for

all n s 2, except when A < 2 ^ in which case the denom­ inator is | A| n“1 .

Proof; As has been noted a consequence of Neumann's work

is that nA is non-abelian indecomposable for all n £ 2.

(a) A further consequence of Neumann's work [12, p. 350]

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is that nA/(hA)' ~ n-1A/(n-1A)1 X (A/A1) for all n a 2. n n n A simple induction argument yields A/( A) ' = x (A/A') i=l for each n a 1. Furthermore it is a simple matter to

show that Z(nA) = Z((n_1A) wr A) =* Z(D) for all n a'2. But Z (D) ~ Z(n_1A) and hence Z(nA) =- Z(A) for all

n a 1. Combining these three results with Corollary

2.10.1 yields |Hj = |A|•|Horn(n-1A,Z(n-1A)jm-1 = | A| •|Hom(n"1A,Z(A)) | m_1

= m* j Horn(n-1A/(n-1A) 1,Z(A))|m-1

= m • | Horn((A/A'),Z(A))| (n“1)(m_1)

for all n a 3. It is noted that the relationship |Horn(A,Z(A))| = |Horn((A/A1),Z(A))| has also been used.

(b) The proof is by induction on n.

n = 2; This is Theorem 2.7 with A = B.

The induction step: Since C2 wr C^ is not a "special

dihedral" group, it follows from Theorem 2.7 that if

n a 3, then

| Aut (nA) | = | nA| -1 1 -1 Aut (n_1A) | -| Aut A|/| A| .|n-1A|

for all groups A. Applying the induction hypothesis

yields |Aut (nA)| =

rai-iHj-ir1*! ai n_;L/( *in_i: -1 Aut a |/

(|a| •|n_1A| ) except when A = in which case the quan­

tity in brackets has denominator |A|n~2 . A simple alge­

braic manipulation gives the desired expression for |Aut (nA)| and completes the induction. #

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For those who wish a perhaps more descriptive formula

in terms of the group A, it is pointed out that a stan­

dard induction argument will yield

|nA| = ml+m+m2+» • •+mn_1 ^ where m = |A ^

The next lemma will be used in the proof of Theorem

2.13 and again in the proof of Theorem 3.2. The lemma

"roughly" stated says: If A^ is a non-trivial indecom­

posable direct factor of the group A and if cp is a homomorphism which acts on A^ like an element of H

would act, Then cp can be extended to a map cp in

Horn(F,F) which is "almost" in the subgroup H of

Aut (A wr B). This lemma would, of course, be even more

significant if the word "almost" could be removed from

the above statement. One way to do that is to impose fur­

ther restrictions on the group A or on both of the

groups A and B. This is exactly what the hypotheses of

Theorems 2.13 and 3.2 do and hence why Lemma 2.12 plays

such an important role in the proofs of these theorems. The precise statement of Lemma 2.12 is:

Lemma 2.12. Let A and B be finite groups. Let s A = x [A.] where A. is non-trivial indecomposable for j=l 3 3 all j = 1,2, • • •, s and let B = { b ^ b ^ .. . ,bn } . Fix

i € {l,2,...,s} and let C = x X [A^] , x€B j^i J L = x [A?] . (Note that F = C x L.) Fix b 6 B and x€ B 1

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let cp £ H o m ^ , AbP) have the property that for each

a 6 A± , 9 (a) = abh for some h in P depending on a. n v, Letting f 6 C and g = n Ha.) k] 6 L define 9 : F -» F k=l K

by cp(f n [ (a,)bk3 ) = f II [cp(a. )bk] . Then cp has the k=l K k=l K following properties:

(a) cp € Horn (F, F)

(b) cp| A. = cp

(c) cp| C is the identity map on C

(d) cp € H if and only if cp is one-to-one or onto.

Proof: First note that cpta^)^ = a^.bbK*.h^b^ where cp(a^) = with h^ g P s Z(F). But bb^ = bbg if

and only if k = s . Hence any two factors in the product

given in the definition of cp commute and the product is unambiguous. It is now possible to proceed with the proof

of (a) through (d) .

(a) Let h^ and h2 be in F and express them uniquely

in the form h 1 = f^g-^ # h 2 = f292 wkere fi'f2 ^ C and

g-, = n [(a, )hk] , g9 = n [(cv )bk] € L. If 9 and 9 1 k=l K ^ k=l K n bv are as in the hypotheses, then since g g0 = n [ (a, c. ) K] 1 2 k=l * k it follows that

9 (hxh2) V ^ ^2^2^ = ^ 1^2^1^2^

= flf2.-J1 I5 (ak°k)blC1

= fif2‘k51t? (ak)blc'iP(ck)bkl •

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Now if 1 £ m,r s n , then

lows that n (ck)bk] = n cp(ak )bk. n cp(ck)bk k= 1 k= 1 k= 1

and therefore cp(h,h_) = f,f0* n cp(a, n cp(c, )bk . 1 2 1 2 k=l ^ k=l * n _ -U But also, n cp (a, ) K £ LP. Hence k=l ^

cpU^h,,) = f1 *kn i9 (ak )bk *f2 *kn ^ ( c k )bk = cpU^) *cp(h2)

and cp € Hom(F,F) . (b) and (c) are clear from the definition of cp and no

proofs will be given.

(d) It is sufficient to show(i) ix°cp -

follows that cpoi(h) = cp(hx) = cp(fX . S (a,)bkX) = fX - ff cp(a.)bkX x k=l K k=l K

= [f- n ip(ak )bk]x = ix (f*kni^(ak )bk) = ix°cp(h) .

Hence ix°cp = cp° ix for all x £ B. To show (ii) it is first recalled that if h € P, then

II (h)*^ = 1 , the identity of F. Now if d £ D, then k=l d = fg where f £ D n C and g has constant value a € . Since cp(a) = a^h for some h 6 P it follows

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. y€BL~ J Hence (ii) is established and the proof is complete. #

Now that Lemma 2.12 is in hand it is possible to give a reasonably complete description of H in the case

that A is a purely non-abelian finite group. The des­

cription is contained in the following theorem and its

first corollary.

Theorem 2.13. Let A be a purely non-abelian finite

group and B a group of order n. Let A = A^ x ••• X Ag where A^. is a non-trivial indecomposable group for each s k = 1 , 2 , . . . , s and let R = x [B x Horn (A, , P) ] . If cp € H k=l K then (a) for each k = 1,2,-••,s there exists a unique b € B such that cp(Ak) <; A^-P . Furthermore, the map.

y:A. -» P defined by y(a) = a-‘1'[cp° j , (a)] for all b a € is in Hom(Ak ,P) . Hence each cp € H determines

in this way an element of R.

(b) The map t:H R defined in (a) is a one-to- one map of H onto R.

Proof; (a) Let k be arbitrary but fixed. Since A

is purely non-abelian, A^ is non-abelian. As was men­ tioned in the proof of Theorem 2.10 , Lemma 2.8 combined

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with the fact that A^. is non-abelian gives the existence

of a unique b € B such that cp (A^ <; Since cp € H

and cp (a) = abf for each a € A^ where f 6 P • it fol­

lows that y(a) = an<^ ;*-t ;'-s straightforward that y

as defined is in HomtA^P) . Hence given cp € H , a pair

(b,y) where b £ B , y € HomtA^P) is obtained for each

k = 1,2,...,s .

(b) Let cp 6 H and let (b,y) be the k— pair in

t(cp) . Then for a € A^ , cp(a) = a^f = a^y (a)'13 and the pair (b,y) determines cp on A^, . Consequently, the s

pairs in t (cp) determine cp on A. But an element of H is completely determined by its action on A. Hence t

is one-to-one. It remains to show that the map t is onto R. Let

r € R / then the following construction will produce a cp € H such that t(cp) = r . For clarity, the proof is

broken into two steps.

Step 1; Let b £ B and y € Hom(Ak ,P). Step 1 shows how

to construct a cp € H such that t(cp) has the ordered

pair (b, y) in the k— position and has the ordered

pair (e,Vj) in all other positions, where e is the

identity of B and v^ is the homomorphism mapping everything in A^ onto the identity of F. The construc­

tion proceeds as follows: Define u £ Horn(A^,A^•P) by u(a) = a^y(a)b for all

a 6 A^ . The map u is a homomorphism since y is a

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homomorphism and the image of y is in P s Z(F) . Now

by Lemma 2.12 u can be extended to a homomorphism cp of

F into F such that cpjA^ = u and cp|c is the identity on C = XXAj where the product is over all x € B and

all j = l,2,***,s such that j ^ k . Since

a 1 [ Cpo j _^(a)] = a 1 Cep (a^* ) ] = a ^ [ cp (a ) ] b -lr , n b-1 -lr b , ,b,b-1 , . = a [u (a) j = a [a y(a) J = y(a) for all a £ A^ , it is clear that if cp is in H, then

t (cp) is the desired s-tuple. Hence by Lemma 2.12 (d)

it remains to show that cp as defined is one-to-one or

onto. It is shown that cp is onto. Clearly C £ imcp

and it is sufficient to show that 5 ;*-mcP • It first necessary to obtain a different description of

x€ B ^ ^k ^ aS a S0t* Define pj bY pj = tf€p = f(B)cZ('Aj)) for all j = 1,2,•••,s . Then P can be regarded as the

direct product of the subgroups P^ , where j = l,2,***,s.

Hence if M = x[Pj] where the product is over all

j = 1,2,--^s such that j / k , then P = M x P^ and any element in P can be expressed uniquely in the form

gh where g € M , h £ P^ • It follows that -> p^

defined by cr = 6 0 y • where 6 is the projection of P onto P^ is in HomtA^P^) . By Lemma 2.9 ,

rise to an automorphism of A^ wr B which is in the sub­

group H of Aut (A^ wr B) . This result is exploited to

yield the description

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. x s V ^ 1 - < *-*gBM f < * » l x ■ £ « * b > 4 1 -

With this different description of as a set» it is now possible to show that cp is indeed onto and hence

in H.

Let f € x(=B[Ak] # f(x) = ax 6 \ for each X in B and V(ax ) = 9xhx where gx € M £ Z(F) and hx € pk •

Then (1) since g”1 € M <; C and cp| C is the identity

on C it follows that cp (g) = g where g = j ^ b ^ x ^ * '

(2) ,p(ax) = u(ax ) = a ^ Y (ax)b , (3) f = X“B U X] ; and

(4) hx =

■P = 9 - ^ B ^ ( ax>b"lxl

" 9-xn€B'ax-v(ax )blb'lx

= g 'x€Btax*Y(ax*]

= g *xeB[ax ^ x " B W ( a x )]X

= •vCR[gvhv.]

*x€BiyxJ 'x€B^hx*

= *-*€b[1£ = *-£Bio(u*)n* •

It is concluded that cp is onto. Hence cp is in H. s Step 2; Since r 6 R , r = x where b^ € B

and Yfc £ HomtAj^P) for all k =1 , 2 , • • • , s . Define for each positive integer n < s the set

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Rn = { r € R : b^ = e and = v^ for all k such that n < k s: s ) and let Rg = R . The proof that t is onto

will be by induction on m. The statement to be proved by

induction is: If A is the direct product of s non- abelian indecomposable factors and m is an integer such

that Isms; s , then for each r 6 Rm there exists a

cp € H such that t(cp) = r . It is noted that Step 1 with k = 1 anchors the induction. Now let s > 1 artd let m be a positive integer such that 2 s m s s. (If s = 1

there is nothing more to show.) Let r € R^ and denote

by the k— ordered pair of r . Hence b^ = e

and = v^ for all k > m. Let r^ be the element of

Rm_^ which agrees with r on the first m - 1 ordered

pairs. By the induction hypothesis there exists a cp^ 6 H

such that t(cp^) = r^ . Since Ym 6 Hom(Am ,P) and

cp1 (P) = P it follows that cp^1"0 vm € Hom(Am ,P) . Construct

as in Step 1 a cp2 in H such that the mr=^ ordered

pair of t(cp2) is and a11 other ordered

pairs are (e,vk ) . Clearly cp = cp1ocp2 is in H. Fur­ thermore, if a £ A^ for some k = 1,2, • , then

cp (a) = cp1oCp2 (a) = ( cp1(a) if k^m

cp1[abm(cp-1oY m (a))bm] if k=m

abk.yk (a)bk if k

a i f k> m

cpx (ab m ) *Ym (a)b m = ab m *Ym (a)b m i f k = m .

Hence t (cp) = r and the proof is complete. #

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33

Recalling that HomtA^P) is isomorphic to the direct

product of |B| - 1 = n - 1 isomorphic copies of Hom(Ak ,Z(A)) yields the following rather explicit formula

for | H| .

Corollary 2.13.1. Under the hypotheses of Theorem 2.13 , | H| =. |B| s.^ni|Hom(Ak/Z(A))|n_1 # where n = | B| .

Remark. If desired, the above corollary can be combined

with Theorem 2.7 on the order of Aut W to give a form­

ula for |Aut W| whenever the group A is purely

non-abelian.

The next two corollaries provide examples of the type

of results which may be extracted from Theorem 2.13 by

considering various special cases.

Corollary 2.13.2. Let A be any finite group such that

Z(A) = {e} and let B be any finite group. Let

A = A^ x *•* X As where is non-trivial indecompos­ able for all k = l,2,...,s . Then H is the direct pro­

duct of s copies of J (B) =- B.

Proof; Since Z(A) = {e} it follows that P = {1) and s hence R is isomorphic to x [B] . Furthermore, an ex- k=l amination of the proof of Theorem 2.13 will show that the

map t; H -» R is an isomorphism of groups when Z (A) = £e 3. The connection with J(B) is emphasized since

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 i f t(cp) = (b1#b2, • • • ,bg) , then cp|A^ = for a11

k = 1,2, •••,s where = { f € F : f(B) = A^) • #

Next recall that Theorem 2.11 dealt with the order

of the subgroup H and the order of Aut W when W was the wreath power of a group A. It is now possible to

give a more complete result in the case that A is a pure­

ly non-abelian group. This result is

Corollary 2.13.3. Let A be a purely non-abelian finite

group. If A = A^ x • • • X Ag where A^ is non-trivial

indecomposable for all k = l,2,***,s , then |Aut(nA)| =

I nA| *| A| s"2. n ' I Hom(Ak ,Z (A) ) | ^2^1 Al . | Aut A| n fnr

all n a 2 .

Proof: Corollary 2.13,1 with A =f= B y^Lqlds | H21 = | A| s. n | Horn (Aj^, Z (A) ) | I Al _1 . The remainder of the

proof is a straightforward calculation using the formulas

given in Theorem 2.11 and is omitted. #

The next three theorems are concerned with the pro- lem of describing the subgroup H of Aut W when A is

abelian or cyclic. Although no definitive results are

obtained about the subgroup H , these theorems show that the problem of determining H in these cases is equiva­

lent to other unsolved problems in group theory.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35

Theorem 2 . 1 4 . Let A be a finite abelian group and B =■ {e,b2, • • • ,bn ) . Let r:H Aut P be defined by

r(

where | ajJ = a prime for each i = l,»**,m apd if S^(P) is the Sylow p^-subgroup of P , then m the kernel of r is isomorphic to x [T.J , where i=l 1 Ti = D n nk . (Si (P) ) for each i = l,..-,m . Furthermore, ker r s Z(H) and the image of r, im r = ( f 5 Aut P :

'Mfb > = f(f)b for all b € B , f 5 P and n [P,j] ¥■

Plj = { 9 6 nk i (Si (P)) : 9b r l = 'X.»(abj*x)} .

Proof: It is trivial that if cp € H , then cp(P) = P.

Hence r is well-defined. By Lemma 2.8 , for each

a € A there exists a b G B and an f G P such that

cp(a) = abf. But A is abelian and hence a**”1 g P . It

can be concluded that for each a in A there exists a

g G P (let g = ab-1f) such that cp (a) = ag .

Now if cp 6 ker r , then cp j P is the identity on P and for each i = l,-»-,m there exists an

fi ^ njc.(si(p)) such that cp(ai) = aifi • Furthermore since a^j”'1’ is in P , a^ -1 = cp(a^_1) = a^?3

for all i and j . Hence for each i > f^j ^ = 1 for all

j and fi G D . Therefore f^ G D n (Si (P)) for m x ^11 i . Now let u:ker r -» x [T.] be the map which i=l 1 associates with epch cp € ker r the ordered m-tuple of

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36

f^'s> determined as above. If tp^ , cp2 € Ker r and

= aifi ' ^2 ^ai^ = ai^i ' t*1611 recalling that gi e D it is seen that (p1ocp2(ai) = cpx (aigi) = cp1(ai)gi =

aifi^i ’ Hence u is a homomorphism. Also, since cp € H is completely determined by its action on the a^

for i = l,**«,m , it follows that u is one-to-rone. m Finally, u is onto, since if (f 1# f2» • • •, fm) 6 X [T.^] ,

the map cp defined by cp(a^) = a^f^ can be extended to

an element cp 6 Ker r such that u(cp) = (f^f*

Hence u is an isomorphism of groups.

To see that ker r £ Z(H) , let cp € Ker r and Y € H . Suppose that cpta^) = aifi and Y (a^) = a^g^^ for

all i = l,**»,m . It is sufficient to show that

cpoY(ai) = Yocp(ai) for all i = 1, ***,m . Noting that

(1) cp| P is the identity on P , (2) f^ £ £ D , and

(3) g± € nk .(P) s P it follows that

cp°Y (a±) = cp(aigi) = cp(ai)gi = a^g.. = a-g.^

= Y(ai)fi = Y t a ^ ) = YoCp(ai). Hence Ker r <: Z(H) .

It is obvious that im r c (Y € Aut P : Y(f^) = Y(f)*5

for all b € 3 , f 6 P and fl [P - •] ^ 0 for all j= 2 1] i = l,...,mj s M . The proof of the other inclusion be­

gins by noting that n [P-•] ^ 0 if and only if j - 2 13 n fi [P-^] is a single coset of T. in n, (S.(P)) . The j_2 1 Ki 1

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 n "if" is clear. Fix i and let g,h € D [P.J.J # then j=2 13 gbj-1 = hb j_1 for all j. Hence (h-1g)b 3 = h'^g for

all j and h ^ g € D . Therefore gT^ = hT^ and the

"only if" is established. Now since A is abelian it

follows that F = P x A and each element of F can be m written canonically in the form f* n (a.) 1 , where f i=l is in P and s^ is an integer for each i . Let Y € M n and choose any g. € fl [ P. . ] . A routine verification j=2 ^ will show that Y*:F -> F defined by

Y* (f * II (a.)Si) = Y (f) • n (a.g.)Si for all f- n (a.)Si<=F i=l i=l i=l is in H and that r(Y*) = Y . Hence M c im r .

Therefore M = im r and the proof is complete. #

A somewhat "sharper" result can be obtained if

(|A|,|B|) = 1. Under these conditions it can be shown

that:

Theorem 2.15. If A and B are finite groups such that

A is abelian and (|a |,|b |) = 1 , then

H { Y € Aut P : Y(fb ) = Y (f.)b for all b e B , f 6 P).

Proof: Let r:H -> Aut P be defined by r(cp) = cp|P for all cp € H . If d 6 D , then bgfiCd(b)] = [d(b) ] I Bl

and it follows immediately from (|a|,|b|) = 1 that

D n P = (l) - Hence it is clear from Theorem 2.14 thet Ker r is trivial and that H is isomorphic to im r .

It is also immediate from Theorem 2.14 that im r is a

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. subgroup of T = ( f g to The proof will be complete if it can be shown that

T c im r . Let Y € T and let i be arbitrary but fixed. Let Y(ab i-1) = f = a ^ _1g. . for each 1 D-; 1 £>-i b • -1 v j = 2,---,n and note that ¥ (e) = 1 . Now ¥ (a^J ) =

¥([a^j_1]x ) = and hence gX . = g^x-g^1

for all j = 2,*--,n . Let t and s be integers such

that 11AI + s | BI = 1 and define g = n (g-u1)3 • It j=2 j follows that

g*'1 = 5 = n <9b .x-g;;Vs- 5 j=2 B1 j = 2 D 1 j = 2 D J X X j=2 D2

- [ nn (g-^)- 5 (gb .)],-[gx],(lBl-1> j=2 b: j=2 b J x

F g, :.[gJs(lBl_1) = [ g J s|Bl = g„ for all x s B .

Hence g € fl [P. •] . But i was arbitrary and it fol- j=2 13 lows from Theorem 2.14 that ¥ € im r . Therefore

T c im r and the proof is complete. #

The final theorem of this chapter is concerned with

the problem of describing the subgroup H of Aut(A wr B)

when A is a finite cyclic group and B is an arbitrary

finite group. It is noted that if A = where |a| = m and B = {b^/b2,* - *,bn ) , then F is isomorphic

to the group ring zm tB] where Zm is the ring of inte­ gers modulo m . The map u:F -» Z [B] defined by

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39

n i • b- n u( n (aJl) x) = 2 [j«b.] gives the desired isomorphism. i=l i=l 1 1 It is this isomorphism which is exploited to yield

Theorem 2.16. Let A = < a) where |a| = m and

B = {b1#b2, • • • ,fc>n) . Let e:Z,n[B] -> Zm be defined by

e(Zjibi) = Sj^ (mod m) and let I = ker e . If

M={l+Q:Q€I and 1 + Q is a unit in zm tB] ) • a subgroup of the multiplicative group of units of the

ring zm [B3 • then t:H -> M defined by t (cp) = u(cp(a)) is an isomorphism of groups.

Proof: Since A is abelian, if cp € H there exists an

n i • b • element g = n (a-31) 1 € P such that cp (a) = ag = i=l n i . b • n -1 a* n (aJi) i and 2 (j •) a 0 mod m . Now v = uocpou i=l i=l 1 is an automorphism of zm tB^ and v (1) = 1 + Q where

Q = Etj^b.^) £ I • Furthermore v(R) = (1 + Q)R for all

R € zm tB] • Clearly vn (l) = (1 + Q)n for all n ;» 1 . Hence since v is in Aut(Zm [B]) there exists a positive

integer k such that vk is the identity on zm tBl and 1 = vk (l) = (1+Q)k . Therefore 1 + Q is in M and

the map t defined by t(cp) = u(cp(a)) = 1 + Q for all

cp g H is into M.

Next let cp-L and cp2 in H and suppose uocp-jQ u-1 (1) = 1 + Q , uocp2ou_1 (1) = 1 + R , then

Uo CPj° ^2° U ^(l) = Uocp-^ou 'L[uocp2oU ^(l)]

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 = U0cp10U_1(l + R) = (1 + Q) • (1 + R) .

Hence t (cp1ocp2) = u (cp1ocp2 (a) ) = uocp^cp^u-1 (1)

= (1 + Q) . (1 + R)

= [uocp1 o U 1 (1) ] * [uocp2 O U 1 (1) ]

= u(cp1 (a)) • u (cp 2 (a) )

= t (cp^) • t (ep2) and t is a homomorphism. It is immediate that ker t is trivial and hence t is

one-to-one.

Now let 1 + Q € M and define v:Zm [B] -» zm [B] by v(R) = (1 + Q)R for all R 6 zm tB] . Note the following

properties of v ; (1) v(R + S) = v(R) + v(S) ,

(2) v (Rb) = (1 + Q) (Rb) = [ (1 + Q)R]b = v (R) b ,

(3) v(kN) = (1 + Q) (kN) = kN + Q(kN) , k € Zm , where n N = E (b.) (note if Q € I , then Q (kN) = 0) , and i=l 1 (4) if (1 + Q)-1 = S , then v(SR) = R for all R in

Zm [B] . Now consider the map u 1ovou:F -» F. Properties

(1) through (4) translate into the following proper­ ties of cp = u ^ovou ; (l) cp is a homomorphism,

(2) cp(fk) = cp(f)*5 for all b £ B , f £ F , (3) ep| D is

the identityon D , and (4) cp is onto (hence one-to-

one) . Hence cp = u-1ovou 6 H and since t(cp) = u(cp(a)) =

u[u ■LoVou(a)] = v°u(a) = v(l) = 1 + Q , the map t is onto.

Combining the above results about t yields the desired result. #

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER III

THE AUTOMORPHISM GROUP OF THE WREATH PRODUCT OF TWO P-GROUPS

Introduction

The study of finite p-groups is an important branch

of group theory. In the case that G is a nilpotent

group, for example, it is well known that G is the di­

rect product of its Sylow p-subgroups. Even when G is

an arbitrary finite group, knowing the structure of the

Sylow p-subgroups often reveals information about the

structure of G . For example, if G is a finite group

whose Sylow p-subgroups are all cyclic, then G has a

normal subgroup N such that (G/N) and N are both

cyclic.

Throughout this chapter A and B will denote fi­

nite non-trivial p-groups. It is noted that if A and

B are finite groups, then W = A wr B is a p-group if

and only if A and B are both p-groups. Hence all

finite p-groups which are also standard wreath products

can be constructed by choosing the p-groups A and B properly.

The most striking result of this chapter is Theorem

3.2 . It gives a one-to-one map of H onto a set R in

a manner similar to that of Theorem 2.13 . The proof

41

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yields a method whereby all the elements of H can be

constructed and a corollary gives the order of H by de­

termining the cardinality of the set R.

The Subgroup H

The first result of this chapter could possibly have

been given as a corollary to Theorem 2.16 . On the other hand, this result will play an important role in the

proof of Theorem 3.2 and therefore could be called a

lemma. However, since it gives a rather complete descrip­

tion of the subgroup H of Aut(A wr B) when A is a

finite cyclic p-group and B is a finite p-group, it has

been decided that its most appropriate position is that of a theorem in this chapter.

Some of the notation of Theorem 2.16 is reviewed in

the hypotheses of the theorem for the convenience of the reader.

Theorem 3.1. Let A = ( a) where |a| = pm and let B = {b^,b2,•♦•,bn } be a p-group (i.e., n = pk for some

integer k) . If e: Z^niCs] -» Z^m is defined by n n n e( S [j,b.]) = E [j-] for all E [j^b.] € Z m[B] and i=l 1 1 i=l 1 i=l 1 1 p I = ker e , then

(a) H =“ 1 + I , a subgroup of the multiplicative group of units of Zpin[B] , and (b) a : Horn (A, P) -» H defined by

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[

Proof; (a) By Theorem 2.16, H^fl + QtQei and

1 + Q is a multiplicative unit in Z^tB] ) . But if

B is a p-group, then I is a nilpotent ideal in Z^rnfE].

It follows that if Q € I there exists a positive integer

k such that Qk = 0 . Therefore,

(1 + Q ) (1 - Q + Q2 - ... ± Qk_1) = 1 + Qk = 1

and 1 + Q is a multiplicative unit in Zpm tB] • I-t can be concluded that H =“ 1 + I . It is noted that the group

of units of Zpm [B] is a direct product of 1 + I and

Zpin where is the group of units of Zpm . A review

of the proof of Theorem 2.16 shows that t: (1 + I) -» H

defined by t(l + Q) = u_1o vo u for all Q € I where v(R) = (1 + Q)R for all R £ ZpmfB] gives the indicated

isomorphism. (b) If f £ Horn(A,P) , then f is completely determined

by its action on "a" , the generator of A. If

T^:Hom(AfP) -» I is defined by T-^f) = u(f(a)) for all

f € Horn(A,P) it is straightforward that is an iso­ morphism of groups where I is considered as an abelian

group under addition. Clearly t2;I *♦ (1 + I) defined by t2 (Q) = 1 + Q for all Q € I is a one-to-one onto map.

It follows that cr = t0T2°Tl : Hom(A'p) H is a one-to-

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one map of Horn(A,P) onto H . But if f 6 Horn(A,P) ,

then or (f) = t.° r 2° t ^ (f) = t° t2 tu (f (a)) 3 = t(l + u(f(a))) = u_1ovou , where v(R) = [1 + u(f(a))]R for all

R € ZpmtB] • Hence o- (f) (a) = u ■L°v°u(a) = u_1°v(l) = u 1[1 + u(f(a))] = af(a) and therefore, since it has already been established that

cr(f) € H ,

or(f) ( n [aji]b i) = n to (f) (a) ji]bi = £ [ (af (a)) ji]bi i=l ' i=l i=l and the proof is complete. #

As has been the practice, whenever possible, a form­

ula is given for.the order of H.

Corollary 3.1.1. If A is a finite cyclic p-group and

B is a finite p-group, then |h| = |A|IBI-1 .

Proof; Since H «** 1 + I , |h | = | 1 + l| = | x| = | P| . #

The next theorem gives a description of the subgroup

H of Aut(A wr B) when A and B are any finite

p-groups.

Theorem 3.2. Let A and B be finite p-groups. Let

A = Cx x *•• X Cr x A 1 x X As where for j = l,»**,r C. = (a.) and I a.I = pxj , x. > 0 and for D D D 3 k = l,»»*,s , A^ is non-abelian indecomposable. Let r R = U x V where U = x [Hom(C.,P)] and s j = l 3 V = x [B x Horn (A, , P) ] . Then if cp 6 H , k=l K

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(a) for each j = l,---,r , CTj:Cj -» P defined by Oj (a”) = (a j [cp (a^ ) ] )n for all n = l,-«-,pX3 is in Hom(Cj,P) and for each k = l,-*-,s thereexists a

unique ordered pair where b^. is that element

of B such that cptA^.) s a^d Y^1^ p is t^ie element of Hom(Ak ,P) defined by Yfc(e-) = a for all a £ A^ , and

(b) the map t:H -» R which associates with each

cp € H the element w $ R as determined in (a) is a

one-to-one map of H onto R .

Proof: This proof follows the pattern of the proof of Theorem 2.13 . The fact that each <7^ is a homomorphism

follows from the fact that a^ 6 Z(F) . The argument that

(b^,y^) satisfies the required conditions is exactly the

same as in the proof of Theorem 2.13 . Consequently, the

map t is well-defined.

Also, noting again that cp 6 H is completely deter­ mined by its action on A , it is straightforward that

cp can be recoved if t(cp) is known. It can be concluded that t is one-to-one.

It remains to show that t is onto. Again, for con­ venience, the proof is broken into two steps.

Step 1. Let b € B and y € HomtA^.P) for some fixed

k. A few minor notational adjustments in Step 1 of the

proof of Theorem 2.13 will show that it is possible to

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construct a cp € H such that t (cp) has the ordered pair

(b,y) in the (r + k) — position and is trivial in all

other positions.

Now if crj € Hom(Cj,P) , then

cp = CTj yields a map cp^ which is in H if and only if cpj is one-to-one or onto. By Lemma 2.12 (c) it is clear

that C = xgB tfj[[Ci] ) x { X [A^] )]X is a subgroup of

im cp j , where A = (is i s r : i ^ j ) . Hence cpj will

be onto if X€B^Cj^X s 'Pj ' Define pi = (f 6 P : f (B) c CL) for all i = l,---,r and

Pk+r = tf 6 P : f (B) = Z(A,J ] for all k = l,...,s . Then P can be regarded as the direct product of the sub­

groups pn / n = 1,2,•••,s+r . Hence if M = n^j(pn] • then P = M x Pj and any element in P can be expressed

uniquely in the form gh where g 6 M and h € Pj • It

follows that p:Cj -* Pj defined by p = 6°cTj where ft is the projection of P onto Pj is in Hom(Cj,Pj) .

By Theorem 3.1 (b) p gives rise to an automorphism of

Cj wr B which is in the subgroup H of Aut (CL wr 3) .

Hence [ p (f (x) ) ] x = f € x|B tCj]x ) . Now

let f 6 xgB^Cj^X ' = ax ^ Cj for eac^ x ^ B and CTj (ax ) = 9xhx where gx € M s Z(F) and hx 6 P. . Then (1) since g”1 € M s C and cpj|c is the identity on C

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47

it follows that cp_j (g) = g where g = X^B (9^1)X »

(2) ¥ j (ax) = ffj(ax) = ax .CT.(ax ) , (3) f = ^ U * ] , and (4) hx = p(ax ) = p(f(x)) for all x € B . Hence

.Pjlgf) = = g-x”B [ax -aj (ax))X

■ 9-X€B[aK)X-x“B [^ < ax>]X - 9-f-x^B[9xV X

" - f-x"B [hxlX

= )) It is concluded that cp^ is onto. Hence cpj € H .

Step 2. Define for each positive integer n < r+s the

set Rn = { w € R : all positions q of w such that

n < q s r+s are trivial ] and let Rr+S = R - rif*:ie proof that t is onto will be by induction. Ihe statement to

be proved is : If A is the direct product of r non­

trivial cyclic factors and s non-abelian indecomposable

factors and m is an integer such that I s m s r+s ,

then for each w € Rm there exists a cp € H such that

t(cp) = w . It is first noted that Step 1 with j = 1 or

Step 1 with k = 1 (if r = 0) anchors the induction.

Now let r+s > 1 (if r+s = 1 the proof is finished)

and let m be such that 2 s m s r+s . Let w € Rm and let w, be the element of R . which agrees with w in 1 m -1 the first m-1 positions. By the induction hypothesis

there exists a cp1 € H such that t (ep^) = w^ .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. case 1; m <; r . Let Yj be the position of w for all j = 1, • • • ,m . Now

the m— position of t(cp2) is 9^ ° Ym anc^ ali other positions are trivial. Clearly cp = cp1°tp2 € H . Furthermore, ( cp1(a) , if a e \ or a € C. , j ^ m cp (a) = cp -j o cp o (a) = ( , ( cp-,^ [ a-cp1 oYm(a)] , if a g Cm

j a*Yj(a) , if a € Cj * jm or a 6 ^

( tPi(a)*Ym (a) = a 'Ym (a) ' if a € cm •

Hence t (cp) = w . case 2: m > r . Let (b ,v ) be the nt^ position of ------m Tm w . Here V i ^ Y m £ Hom(Am_r'p) and ifc is possible to construct as in Step 1 a cp2 € H such that the im­

position of M

ment similar to that in case 1 will show that t(cp) = w .

Hence in either case t(cp) = w and the proof is complete. #

As usual, a formula for |H| is determined from the cardinality of the set R . The result here is

Corollary 3.2.1. If A and B are finite p-groups, then

I Hi = IT |Hom(C.,Z(A) ) I lBl-1.|B| S. n I Horn (A. , Z (A) ) I I Bl _1 j=l 3 k=l k

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49

where A has the structure indicated in Theorem 3.2 .

Proof: P is isomorphic to the direct product of |B|-1

isomorhic copies of Z(A) . #

Before continuing the list of other corollaries and

remarks which can be gleaned from Theorem 3.2 , a slight

digression seems appropriate.

It is noted that the map t:H -> R given in Theorem 3.2 exists for any finite groups A and B since its

definition did not in any way depend on the fact that A

and B were p-groups. Furthermore, even in the more

general situation t is still one-to-one. It follows

that if A and B are finite groups, then | H| <; |R| .

A lower bound for |h| can also be obtained. The con­

struction found in Theorem 2.13 can still be used for

the non-abelian direct factors of A. Hence it can be

concluded that {(1,1,•••,1)] x V c im t . Furthermore if

C = (a) is a direct factor of A , then cr:C -» C-P de­ fined by or (a11) = (a11)*5 for any fixed b € B will extend

by Lemma 2.12 to an element of H . This shows that at

least part of U x [(e,v]L),— , (e,vg) } is always in

im t where v^ is the trivial homomorphism. These ob­

servations are condensed into the following remark.

Remark. Let A and B be finite groups. If

A = C1 x ••• X Cr x X ••• X A g where C.. = (a..) and

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|aj| = (Pj)X3 for some prime p^ and integer x.. > 0 and A^ is non-abelian indecomposable for each

k = l,***,s , then if n = | B| ,

n r+s-^n jHomtA^P) |£|h|s n |Hom(Cj,P) | *ns *^n J HomfA^P) | .

Furthermore, these bounds are best possible.

Proof: The bounds are obtained as mentioned previously.

In case both A and B are finite p-groups. Corollary

3 2.1 shows that the upper bound is attained. When A is purely non-abelian the upper and lower bounds coincide.

An examination of W = wr C2 will show that :the bounds

given above are 2 s |H| s 3 and in fact |h| = 2. It

follows that the given bounds are best possible. #

It is further noted, without proof, that |B|r+s

divides |H| for any finite groups A and B where A

is decomposed as above. The digression having been completed, A and B

will resume their role as finite p-groups for the rest

of this chapter. Also, as mentioned earlier there are other corollaries and remarks which can be obtained from

Theorem 3.2 . The next one of these to be considered is

a result which was actually first proved by using Lemma

2.8 and a well-known result on p '-automorphisms of p-groups. However, it now seems appropriate to include

it in this thesis as a corollary to Theorem 3.2 .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51

Corollary 3.2.2. If A and B are finite p-groups, then

H is a p-group.

Proof: Since A is a p-group all of the groups of homo- morphisms appearing in the formula for |H| given in

Corollary 3.2.1 are p-groups. #

Combining Theorem 3.2 with Theorem 2.7 on the order

of Aut W yields the next corollary. It is pointed out

that Khoroshevskii [10, p. 56] has proved the "if" por­

tion of this corollary for p an odd prime. The author

is indebted to Khoroshevskii since it was his paper which

suggested Lemma 2.8 .

Corollary 3.2.3. If A and B are finite p-groups, then

Aut W is a p-group if and only if Aut A and Aut B

are p-groups.

The next corollary gives a formula for the order of

Aut W when A and B are both abelian p-groups. The

formula is complete in the sense that if the abelian groups A and B are known, then |Horn(A,A)| , |Aut A| .

and |Aut B| can be computed from well-known formulas.

See, for example, Sanders [15, p. 226] .

Corollary 3.2.4. If A and B are finite abelian

p-groups, then |h | = | Hom(A,A)|IB l a n d hence

( 8 , if A B “■ C 0 | Aut W| = ( . . ( |A|I I *|H|*|Aut A|•|Aut B| , otherwise.

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Returning to the idea of the wreath power of a group

A , the following three remarks are made. They are all

immediate consequences of Theorem 2.11 , the corollaries

of Theorem 3.2 , and some simple algebraic manipulations.

Hence no proofs are given.

Remark 1. If A is a finite p-group, then for n 2 2 2 (2"+(§)-2) if

| A| • n |Hk |.|Aut A | /| A | , otherwise

where |H2| is given by Corollary 3.2.1 with A = B and

IHjJ = | A| • | Horn((A/A1 ) ,Z(A)) | (k-1) (l Al _1) for all k * 3 .

Remark 2. If A is an abelian p-group and A / (see

Remark 1) then |Aut(nA)| = |nA| • | Hom(A, A) | (2J (l Al _1) . j Aut A|n/| A| 2

where if m = |A| , then In,| 1+m+m +•••+m I A| = m

A special case of the first two remarks is

Remark 3. If A = Cp , p an odd prime (for p = 2 see

Remark 1) , then

|Aut(nC )| = |nC |.p*2} (p_1).(p-l)n/p2 for n a 2 .

the permutation group on p symbols. Hence the above

remark gives the order of the automorphism group of that

group.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Some Results on the Structure of Aut W

Some information can be obtained about the structure

of a Sylow p-subgroup of the outer automorphism group of

W. This information is given in Theorem 3.4 where A

and B are finite p-groups and in Theorem 3.5 where B

is further assumed to be abelian. First however it is

convenient to describe a Sylow p-subgroup of Aut W. The

description that will be found to be most useful is given in the following lemma.

Lemma 3.3. Let A and B be finite p-groups except that

if B = C„ , then A / C„. If A and B are Sylow 2 2 p p p-subgroups of Aut A and Aut B respectively, then

Ap* and Bp* are Sylow p-subgroups of A* and B* re­

spectively. Furthermore, Q = A ^ H I ^ ^ * is a Sylow

p-subgroup of Aut W and is a splitting extension of

Ap'*HI1 ^ Bp * .

Proof; Since Aut A A* and Aut B “* B* it is immedi­

ate that Ap* and Bp* are Sylow p-subgroups of A* , B*

respectively.

Now recall that HI.^ is normal in Aut W and that

A* and B* commute elementwise. Since HI.^ is a

p-group it follows that HI^ e N is a normal p-subgroup

of Aut W . Further recalling that Aut W is a splitting

extension of A*N by B* it is clear that A *NB * is P P 53

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a p-subgroup of Aut W and a splitting extension of A *N by B * . It remains to show that the index of P P Ap*NBp* in Aut W is relatively prime to p . Using

the fact that Q is a splitting extension it is immediate

that it is necessary to show that |A*N:Ap*N| = |A*:Ap*| .

But N is a normal p-subgroup of Aut W . Therefore A* f| N is a normal p-subgroup of A* and hence A* 0 N =

Ap* f| N . A simple order argument yields the desired

result. #

It is now possible to give information about the

structure of a Sylow p-subgroup of Out W .

Theorem 3.4. Let Aand B be finite p-groups except

that if B = C2 , then A / C2 . If Q is as defined in

Lemma 3.3 , then R = Q/(Inn W) is a Sylow p-subgroup

of Out W . If S = Ap*(Inn W)/(Inn W) and T = HBp*(Inn W)/(Inn W) , then R is a splitting exten­

sion of T by S . Furthermore, S is isomorphic to a

Sylow p-subgroup of Out A and T Bp*H/l(B) where

1(B) = (ib € Inn W : b € B) .

Proof: Since 1^ s Inn W it is immediate that R = ST .

If t e S n T , then t = a*(Inn W) where <** 6 Ap* and

t = hp*(Inn W) where h € H , P* € Bp* . It follows

that a * = for some x 6 B , f € F . Now if a * € Ap* , then b = a*(b) = hp*ix f (b) for

all b 6 B. But h_1|B is the identity on B and

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(p*) ^(b) = p— (Id) for all b € B . It follows that p”1 ^) = ix f (b) = if (bx) G B for all b G B . Hence

f € F n N(B) = D . Therefore p-1 (b) = if (bx) = bX =

ix (b) for all b G B and p* = :(i -1)* • It is a rou­

tine exercise to show that i^ = ix*jx f°r x 6 B

where G J(B) £ H . Also since f G D , ixf = ifix =

ixif and it follows that hp*ixf = 11 ^x -1^ *ixif =

h(ix-l)*ix*jxif = h ^xif * Also note that if f G D and f(x) = a for all x G B , then i^ = ia* . Combining

the above a * = hp*ixf = hjxia* and hence a*(ia-l)* =

hjx € A* n H = {1) . It follows that a * = ia* € Inn W and hence S f| T = {1} . Since HI1 is normal in Aut W and A* , B* com­

mute elementwise it is straightforward that T is nor­

mal in R and hence R is a splitting extension of T by S . Next Ap* s Ap*(Inn W) , Inn W is normal in Ap*(Inn W) and Ap* n Inn W = (Inn A)* . Hence

S = Ap*(Inn W)/(Inn W) =- Ap*/(Inn A)* which is a Sylow

p-subgroup of Out A * Similarly, HBp* £ HBp*(Inn W) , Inn W is normal in HBp*(Inn W) and HBp* n Inn W = 1(B),

hence' T = HBp *(Inn W)/(Inn W) =- HBp*/l(B) . #

If B is abelian a Sylow p-subgroup of Out W can hlso be viewed as a splitting extension of

Ap*H(Inn W)/(Inn W) by Bp*(Inn W)/(Inn W) . It is noted

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that this splitting is compatible with the splitting of Aut W . Even more information about the situation when

B is abelian is given by the following theorem.

Theorem 3.5. Let A and B be finite p-groups with B

abelian except that if B = > then A / C2 . If A^*

and Bp* are Sylow p-subgroups of A* and B* respec­

tively , S = Ap*(Inn W)/(Inn W) , U = H(Inn W)/(Inn W) ,

and V = Bp*(Inn W)/(Inn W) , then Y = Ap*H(Inn W)/(inn W)

is a splitting extension of U by S and R = Ap*HI^Bp*/(Inn W) , a Sylow p-subgroup of Out W is

a splitting extension of Y by V . Furthermore, S is

isomorphic to a Sylow p-subgroup of Out A , U =* H/l(B) ,

and V B * . P

Proof; It is clear that Y = US and an argument similar

to that of Theorem 3.4 will yield U n S = Cl) . (Note

that J (B) = 1(B) H .) Furthermore, H is normal in A*H and it follows that Y is a splitting extension of

U by S . The fact that R is a splitting extension of Y by V follows from (1) J(B) = 1(B) s H and (2) Aut W

is a splitting extension of A*HI^ by B* .

As before, S = Ap*/(Inn A)* . Also H s; H(Inn W) ,

Inn W is normal in H(Inn W) and H n Inn W = 1(B) .

It follows that U ~ H/1(B) .

Finally, Bp* £ Bp*(Inn W) , Inn W is normal in

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B *(Inn W) and B * n Inn W = {l) since 0* = i ,= leads P P xr to the conclusion that p*(b) = if (b) = b(f 1)bf for all

b £ B and hence that p* = 1 . It is concluded that

V = Bp*(Inn W)/ (Inn W) ~ Bp* . #

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER IV

THE CONJECTURE "|g | DIVIDES |AUT G|"

Introduction

The conjecture to be considered in this chapter is: 2 If G is a non-cyclic p-group of order greater than p ,

then |G| divides |Aut G| .

There has been some interest in this conjecture in

recent years. The conjecture has been established for

p-groups of class 2 [5] , for p-abelian p-groups [3] ,

and for other classes of p-groups [2] , [4] . Inroads

into the general conjecture however, have been slight.

The best known result in this direction seems to be a re­

sult of Gaschutz [6] , [7] (for a very readable account

of this see Gruenberg [8, p. 110]) which states: If G

is a finite p-group of order greater than p , then p

divides |Out G| . It is noted that the conjecture states

that |Z(G)| divides |Out G| . The results of this chapter will extend the classes

of groups for which the conjecture is known to be true.

The main results are Theorems 4.3 and 4.5 .

G as a Central Product

Otto [11, p. 282] has shown that if a p-group

G = A x B where A is abelian, B is purely non-abelian

58

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and o(B)|o(Aut B) , then o(G)|o(Aut G) . A result of

this type not only extends the number of groups for which

the conjecture is known to be true, but also, and perhaps

more importantly, shows that the truth of the overall con­ jecture depends only on being able to prove it for a

smaller . lass of groups. Otto's result shows that it is sufficient to consider p-groups with no non-trivial abel­

ian direct factors, i.e., purely non-abelian p-groups.

The main result of this section is Theorem 4.3 . It

is noted that Otto's result is a special case of this

theorem. The first result of this chapter is concerned with

the ability to extend an automorphism of a particular

subgroup of a group to an automorphism of the entire group. The lemma is of some interest in its own right.

Lemma 4.1. If G =

trivial maximal subgroup of the p-group G , then

a g Aut H extends to a 6 Aut G if and only if ot (ap) =

asphp where 0 < s < p and h g Z(H) .

Proof; Let a € Aut H be extendible to a € Aut G .

Noting that each element g g G can be expressed uniquely

in the form g = a^h , 0 s k < p , h g H , it is seen that a (a) = ash , Oss

otherwise ot would map G into H and not be onto.

Also, a g Z(G) implies a(a) = ash g Z(G) and so

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h € H f| Z (G) = Z(H) . The "only if" now follows easily. Conversely, let ot € Aut H and suppose ot (a^*) = asphp , 0

(h') for each g = a^h* 6 G . To see that ot is

a homomorphism let 9 ^ 9 2 € G • Then = aklhx , g2 = ak2h2 where 0 s k.^,^ < p , h^,h2 6 K . Let

k l + k2 = k3 + rp where 0 ^ k3 < P a n d r s; 0 . It follows that a (g1g2) = a (ak'3arph1h2)

= ask3hk3<* (arp)<* (h1)cv (h2) = ask3hk 3asrphrpa (h-^a (h2)

= as(k1+k2)hk1+k2a(hi)a(h2) = aaklhklff(hi)ask2hk2a(h2)

= a(g-L)a(g2) • Clearly a extends a 7 furthermore

a(aklh1) = Q'(ak2h2) implies asklhkla (h1) = ask2hk2ar (h2)

and hence as k 2^ € H . Therefore k^ = k 2 . It fol­

lows that ot (h^ = a(h2) and hence hx = h 2 . Hence ot

is one-to-one and ot € Aut G . #

Theorem 4.3 can be obtained quite easily from the

following special case. Before stating the next lemma

however it is necessary to make the following definition.

If G is a finite p-group, then ^(G) = < ( g € G : o(g) |pk }> .

Lemma 4.2. If G =

ial maximal subgroup of the p-group G and o(H)|o(Aut H),

then o (G) | o (AutjjG) | o (Aut G) .

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Proof: The proof is broken into two cases. case 1. o(a) = p . This is actually a special case of

Otto's result. However, an independent proof is given so

that Otto's result might be seen to be a corollary of

Theorem 4.3 . It is first noted that every automorphism of H ex­

tends to an automorphism of G. Hence the restriction

map 0:AutHG -> Aut H is onto.

Let of 6 ker e . It follows that a (a) = ash where 0 < s < p and h 0 Q1 (Z(H)) . Conversely, for each s

such that 0 < s < p and h 0 n1 (Z(H)) define 0:G -> G

(dependent on the choices of s and h) by

gfa1^) = as'*ch'*ch^L for all alch1 0 G . It is routine that

0 0 ker 0 . Hence o(ker 0) = (p - 1)*o(n1 (Z(H))) . It

can be concluded that otAutjjG) = o (ker 0)-o(Aut H) =

(p - 1) *o (nx (Z (H))) *o (Aut H) . Since o(H)|o(AutH) it

follows that o(G)|o(AutHG)|o(Aut G) .

case 2. o(a) > p . A restatement of Lemma 4.1 shows

that a 0 Aut H extends to a 0 Aut G if and only if the

induced automorphism a * 0 Aut[Z(H)/$(Z(H))] leaves

(a*5) ^ Z (H)/$ (Z (H) ) invariant. Consider the permutation

representation (Aut H,X) ; where X = (cyclic subgroups

of Z(H)/ $ (Z(H))] . Then E = ( a € Aut H : a is extend­ able to a 0 Aut G) = St((ap>) , the stabilizer of (aP). Now |Aut H : E| is the order of one orbit and hence

| Aut H : E| <; |X| = (pr - l)/(p - 1) < pr where

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Z (H) /$ (Z(H)) is elementary abelian of rank r . Hence pr > (|Aut H|p )/(|E|p ) and (| Aut H| p) | pr-1* | E|p where

|G|p is the highest power of p dividing |G| . Using

the hypothesis that o(H)|o(Aut H) it is seen that

o(G)|pr -|E|p and the proof will be complete if

pr*(|E|p )|o(AutHG) . Again consider the restriction map

0: AutjjG -> Aut H and note that im Q = E . Let a 6 ker 0

and suppose that a(a) = ash where 0 < s < p- and

h 0 Z(H) . Then ot (ap) = [o'(a)]p = ap implies

aSh s a mod n (Z(G)) . Clearly, (Z(H)) £ (Z(G)) . If this inclusion if proper, there exists an element

a 1 0 Z(G) with a 1 H and o(a') = p . Hence case 1

applies. Therefore without loss of generality

fJ1(Z(H)) = n (Z (G)) and consequently ash = ah' for some

h' 0 nl (Z(H)) . Conversely for each h' 0 n1 (Z(H)) the

map ot - . G -» G defined by a ( a^h) = (ah')^h for all a1^ 0 G is in ker 0 . It follows that o (ker 0) =

o(ni (Z(H))) = o(Z(H)/*(Z(H))) = pr . But AutjjG/ker 0 ~ E and hence pr -o(E) = o(Aut^G) . This implies pr -(|E|p ) |o(AutHG) completing the proof. #

It is now possible to state and prove the main result

of this section.

Theorem 4.3. If a p-group G is the central product of non-trivial subgroups H and A , where A is abelian

and o(H)|o(Aut H) , then o(G)|o(Aut G) .

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Proof: Note that A s Z(G) and let |A : H n A| = pn

where n > 1 . The proof is by induction on n .

n = 1. Lemma 4.2 gives this result for without loss of

generality A = (a) . The induction step. Let n > 1 and select A^ s A such

that H n A < A1 < A and |A ± : H n A| = p . Let = HA1 . It follows that H n Ax = H n A and

|A1 : H n A1 | = p . Hence by Lemma 4.2 o(H1)|o(Aut H1) .

But G = H1A and ^ fl A = Ax . Hence | A : H1 D A| =

pn 1 and the induction hypothesis yields o(G)|o(Aut G). #

As was noted earlier, Otto's result shows that the

truth of the overall conjecture depends on being able to prove it for purely non-abelian p-groups. Theorem 4.3

has a similar effect as can be noted from the following

remark. The proof of this remark is straightforward and

hence is omitted.

Remark. For a finite p-group G , the following are

equivalent:

(i) G is the central product of proper subgroups

A and H where A is abelian.

(ii) There exists a maximal subgroup K of G such

that Z(K) < Z(G) .

(iii) Z(G) / $(G) .

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. G as a Wreath Product

This section investigates the conjecture "o(W) di­ vides o(Aut W)" when the group W is a standard wreath

product of finite p-groups. Many of the results of Chap­

ters II and III can be used to shed some light on the con­ jecture in this case. The first theorem to be stated can

be seen to be of the same type as Theorem 4.3 . This

theorem has a number of consequences, one of them is (for­

tunately or unfortunately?) that standard wreath products

are not the place to look for counterexamples to the

conjecture.

Theorem 4.4. If A and B are finite p-groups,

W = A wr B and o(A)|o(Aut A) , then o(W)|o(Aut W) .

Proof: The proof is in two cases. case 1. A = B = C2 ., Here W = DQ “ Aut Dg . Clearly,

o (W) | o (Aut W) .

case 2. Not both A and B are C2 By Theorem 2.7 o(Aut W) = o(W)*o(H)-o(Aut A).o(Aut B)/o(A)*o(B) . Since

o(A)|o(Aut A) it is sufficient to show that o(B) divides

o(H)*o(Aut B) . But Lemma 2.6 shows that J(B) £ H and

J(B) “■ B . It follows that o (B) | o (H) | o (H) *o (Aut B) . #

An interesting corollary to Theorem 4.4 is

Corollary 4.4.1. If A is a finite abelian p-group, B

64

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is a finite p-group and W = A wr B , then o(W)|o(Aut W) .

Proof: The proof is divided into four cases. 2 case 1. A non-cyclic and o(A) > p . It is well known

that under these conditions o(A)|o(Aut A) . Hence

Theorem 4.4 yields the result in this case. case 2. A =- x . It follows from Corollary 3.2.1

that |H| = |A|1Bl“1 -|A|IB l-1 . A simple algebraic argu­

ment will show |h| = |A|2 ^IB I-'L^ = p ^ l B l ^ p> -1B | .

But H is a p-group and hence p -|B| divides |H| .

Since p| o(Aut A) it follows that o(A)-o(B) divides

o(H)-o(Aut A) . Theorem 2.7 now yields the desired

result.

case 3. A cyclic, except if B = > then A ^ .

If o(A) = pn , then o(Aut A) = pn_1(p - 1) • In this

case Corollary 3.2.1 gives |H| = |a|IB I-1 and another

algebraic argument will show that |h | = |A|IB I ^ s p| B| .

Hence again o(A)*o(B)|o(H)»o(Aut A) and case 3 can be

established.

case 4. A = B = C2 . Again C2 wr C2 Dg “ Aut DQ . #

Corollary 3.2.1 again plays an important role in the

following theorem.

Theorem 4.5. Let A and B be finite p-groups and let

A = C x A^ x • • • X Ag where C is abelian and A^ is non-abelian indecomposable for all k = l,**«,s . If

W = A wr B and 1^1 divides

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iHomfA^Z (A^)) | lBl -1-1 Aut for all k = l , . . . , s ,

then o(W)|o(Aut W) .

Proof; There are two cases to consider. case 1. s=0. If s = 0 , then A is abelian and

Corollary 4.4.1 yields the desired result.

case 2. s>0. If s > 0 , then

O (Aut W) = O (W) *0 (H) -O (Aut A) • o (Aut B)/o(A)*o(B) .

Define Pk = { f € P : f(B) c Z(A]c)) , then

| Horn (A^, Z (A^,) ) | I Bl “1 = | Horn (A^/P^) | which divides s | Horn (Aj^P) | . It follows that n | A^.| divides

s | Aut A| • n |Horn(A^,P)| . Next suppose without loss of

generality that C = C1 x ••• X cr where is cyclic of order pXr and x^ s x2 ^ ••• s xr . It is immediate

that | C| divides |Hom(Cr,P)| . It can be concluded r that |C| divides n |Hom(C.,P)| . j=l 3 Now Corollary 3.2.1 gives the order of H as

r q s I H | = n |Hom(C. ,P) | -1 BI - n | Horn (A, , P) | . Hence j=l 3 k=l o(A)*o(B)|o(H)*o(Aut A) and the result follows. #

■Theorem 4.5 can be used to obtain a number of con­

ditions on the p-groups A and B which would guarentee

that o(W)|o(Aut W) . An example of one such condition is

given in the following corollary.

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Corollary 4.5.1. If A and B are finite p-groups and

| B | s | Z (A) | , then o(W)|o(AutW) .

Proof; If A is abelian there is nothing to show; hence

let A^ be a non-abelian indecomposable direct factor of

A . Since 2pm s 2m+1 s m + 2 for all integers m s 1

and all primes p , it follows that (p2) ;> |b | s

| Z (A) | s | Z (A^) | . Now |Hom(Ak#Z(A]c))| = | Horn((A^/A^),Z(A^))| and so p2 divides |Horn(A^,Z(A^))| .

Therefore j Z (A^) j <; (p2)^lB l-1^ | Horn (A^ Z (Ak) ) | I Bl “1

and it can be concluded that |Z(A^)| divides

|Horn(A^,Z(A^))|IB l_1 . It is immediate that |A^| divides

| Horn (A^, Z (A^ ) | I Bl -1* | Aut A^| . Theorem 4.5 now yields

the desired result. #

An almost trivial consequence of Corollary 4.5.1 is

contained in the following

Remark. If A is a finite non-trivial p-group, then

o(nA)|o(Aut[nA]) for all n ^ 2 .

Proof: Z(nA) =* Z (A) for all n ;> 1 . Hence |a | s

|z(nA)| for all n a 1 . #

Although the previous theorems, corollaries, and re­

mark have shown that o(W)|o(Aut W) under many different

and varied hypotheses on the finite p-groups A and B ,

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no proof has yet been given to show that it is always

true. In fact, no proof has yet been found by this

author. One possible approach to the more general result

is indicated by the following theorem. This theorem shows

that the conjecture is true for the wreath product of any

two finite p-groups if a weaker result is true for all

non-abelian indecomposable p-groups.

Theorem 4.6. The following Statements are equivalent. (i) If A and B are finite non-trivial p-groups

and W = A wr B , then o(W)|o(Aut W).

(ii) If G is a non-abelian indecomposable

p-group, then o(G)jo(Horn(G,Z(G))| P ^-ofAut G) .

Proof: (i) implies (ii) . Let G be a non-abelian

indecomposable p-group and consider W = G wr Cp . Now

by Theorem 2.7

o(Aut W) = o(W)-o(H)-o(Aut G)-o(Aut Cp )/o(G)-o(Cp )

and by Corollary 3.2.1 , o(H) = p*o (Horn (G, P) ) = p.o(Hom(G,Z(G) ) )P_1 .

It follows that o(G)|o(Horn(G,Z(G)))p—1.o(Aut G) since

it has been assumed that o(W)|0(Aut W) . (ii) implies (i) . If (ii) is true, then di­

vides o(Horn(A^,Z(A^)) ) -o(Aut A^) for all non-abelian indecomposable direct factors A^ of A. Theorem 4.5

now yields the desired result. #

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The thesis is concluded with the following remark.

The reader is asked to notice the implications of this

remark as it pertains to Theorem 4.6 and Theorem 4.5 .

Remark. If G is a non-abelian indecomposable finite

p-group and exp(Z(G)) s exp(G/G’) , then

o(G)|o(Horn(g ,Z(G)))• o (Inn G) | o(Horn(G,Z(G)))p-1.o(Aut G)

Proof; O (Hom (G, Z (G) ) ) = O (Horn ( [G/ G' ] , Z (G) ) ) 2: O (Z (G) )

and hence o(Z(G))|o(Hom(G,Z(G))). The result is now

immediate. #

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. BIBLIOGRAPHY

1. Adney, J.E. and Yen, Ti, "Automorphisms of a p- group." Illinois J. Math., 9 (1965), 137-143.

2. Davitt, R.M., "The Automorphism Group of a Finite Metacyclic p-group." Proc. Amer. Math. Soc., 25 (August 1970), 876-9.

3. Davitt, R.M., "The Automorphism Group of Finite p-abelian p-groups." Illinois J. Math., 16 (March 1972), 76-85. 4. Davitt, R.M. and Otto, A.D., "On the Automorphism Group of a Finite p-group with the Central Quotient Metacyclic." Proc. Amer. Math. Soc., 30 (November 1971), 467-472.

5. Faudree, R., "A Note on the Automorphism Group as a p-group." Proc. Amer. Math. Soc., 4 (1953), 15-26.

6 . Gaschutz, W., "Kohomolische Trivialitaten und aussere Automorphismen von p-Gruppen." Math. Zeit., 88 (1965), 432-3.

7. Gaschutz, W., "Nichtabelsche p-Gruppen besitzen ciussere p-Automorphismen." J_;_ of Algebra, 4 (1966), 1-2 .

8 . Gruenberg, K.W., Cohomoloqical Topicd in Group Theory. Lecture Notes in Mathematics no. 143. Springer- Verlag, Berlin, 1970. Pp. xiv + 275.

9. Houghton, C.H., "On the Automorphism Group of Certain Wreath Products." Pub. Math. Deb., 9 (1962), 307-313.

10. Khoroshevslcii, M.V., "On the Automorphism Groups of Finite p-groups." Algebra and Logic, 10 (January- February 1971), 54-7.

11. Neumann, B.H. and Neumann, Hanna, "Embedding Theorems for Groups." London Math. Soc., 34 (October 1959), 465-479.

12. Neumann, P., "On the Structure of Standard Wreath Products of Groups." Math. Zeit., 84 (1964), 343-373.

13. Otto, A.D., "Central Automorphisms of a Finite p- group." Trans. Amer. Math. Soc., 125 (1966), 280-7.

70

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14. Ree, R., "The Existence of Outer Automorphisms of Some Groups. II." Proc. Amer. Math. Soc., 9 (1958), 105-9. 15. Sanders, P.R., "The Central Automorphisms of a Fi­ nite Group." J. London Math. Soc., 44 (1969), 225-8.

16. Schmidt, O.U., Abstract Theory of Groups. (ed. J.B. Roberts) San Francisco: W.H. Freeman & Co., 1966. Pp. vi + 174. 17. Scott, W.R., Group Theory. Englewood Cliffs, N.J.: Prentice-Hall, 1964. Pp. ix + 479.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.