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ABSTRACT
SOME STABILITY GROUPS OF FINITE GROUPS
By

Allen Lee Bertelsen

A topic of study in finite group theory is the group of auto-
morphisms of a finite group. One method for studying automorphisms
is to look at their effect on chains of subgroups, rather than
individual elements.

Given a chain of subgroups s: G = G 2 G 2...2 Gn = l, we

0 1
define Stab(s) by

)a=gG

Stab(S) = {0! 5 A“ G\(giGi+l i i+l

for all gi E Gi’ i = 0,1,2,...,n-l].

P. Hall has shown that Stab(s) is a nilpotent group of class less
than or equal to (2).
If A s Stab(s), there is a canonical chain

S: G 2 [G,A] 2 [G,A,A] 2...2 l in G and we define the closure of

A, written A; by
K’= Stab(s)

A is said to be a closed stability group if AI= A.
In Chapter I we have:
(1) K = K’

(2) The prime divisors of A are the same as the prime

divisors of A.

Allen Lee Bertelsen

(3) NAut G(A) = {a t Aut G‘B leaves each group in s invariant}.

(4) If A <1Aut c, then AdAut c.

Two questions arise
(1) Which nilpotent groups are stability groups?
(2) Which stability groups are closed?

Using the following

If A S Z(G), the center of G, then

Hom(G/A, A) Q'Stab(G 2 A 2 1)

f
feaf-g-‘gg
we showed that

Any abelian group is a closed stability group.

The search for stability groups may be simplified by

A s Aut G is a closed stability group if and only if for every

p dividing \A\, a p-Sylow of A is a closed stability group.

Let G=HXK and

s: H x K 2 H x K 2...2 H X K = l x K 2 K 2...2 K = 1
1 s 1 n

then Stab(s) is the semidirect product of Stab(H X K 2 K 2 l)

and a subgroup isomorphic to Stab(H 2 H 2...2 l) x Stab(K 2 K 2...2 1).

1 1

Let G be nilpotent and h the isomorphism from Aut G
onto the direct product of the automorphism groups of the p-Sylows
of G. A s Aut G is a stability group if and only if

A" = n Stab(s ) where s is a chain from the p-Sylow of G to 1.
FUN p p

Allen Lee Bertelsen

This last theorem leads us to Chapter II and stability groups
of p-groups which must be p-groups.
If G is a p-group then:
(1) Any p-Sylow of Aut G is a closed stability group.
(2) Op(Aut G) is a closed stability group.
If G is a p-group with G' 5 2(6) or \Z(G)\ = p, then
Stab(G 2 Z 2 l) is the group of central automorphisms.
If G is a p-group and A S Aut G‘ is of the form
(i) A is a p-groupo
(ii) A is normal in every p-Sylow of Aut G that
contains. A.
(iii) A is the intersection of all p-Sylows of Aut G
that contain A.
then A is said to be of K—type.
If G is a p-group and A S.Aut G is of K-type then A
is a closed stability group.
Let G be an elementary abelian p-group. Then
(1) (Kaloujnine) A is a stability group if and only if
A is of K-type.
(2) A is a minimal stability group if and only if
A = Stab(G 2 H 2 l) for a subgroup H of G. A
minimal stability group is one that contains no other
nontrivial stability groups.
(3) If A and B are two stability groups in the same
p-Sylow of Aut G, then <A,B> = AB is a stability group.
(4) Stab(G 2 G1 2...2 Gn = 1) is the product of the minimal

stability groups Stab(G 2 Ci 2 l), i = l,2,...,n-l.

Allen Lee Bertelsen

If G is a p-group for which every stability group is
Kaloujnine, then G is elementary abelian or cyclic of order p2.

In Chapter III we examine Fitt(Hol G), the Fitting sub-
group of the holomorph of G, and Fitt(Aut G), the Fitting sub-
group of Aut G.

If A is the product of all stability groups of characteristic
series of G then Fitt(Hol G) = A-Fitt G.

If G is a p-group, not 0(2) X 0(2) or 0(3) X 0(3), then:

r
Op(Aut G) when G is nonabelian or

Fitt(Aut G) =
Op(Aut G) X B where B is a cyclic subgroup of

 

Kthe center of Aut G and \B\ = p-l.

If Fitt G is purely nonabelian or if Exp Z(G) divides
Exp(c/zc') then Fitt(Aut G) is a closed stability group.

In Chapter IV we have:

The quaternion group of order eight is a closed stability
group but is not a closed stability group for a normal series in a

2-group.

SOME STABILITY GROUPS OF FINITE GROUPS

By

Allen Lee Bertelsen

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1974

ACKNOWLEDGMENTS

I would like to thank.Dr. J.E. Adney for his patience and

guidance in the preparation of this thesis.

ii

Chapter

I

II
III

IV

TABLE OF CONTENTS

INTRODUCTION

DEFINITIONS AND ELEMENTARY PROPERTIES OF
STABILITY GROUPS AND CLOSED STABILITY GROUPS

STABILITY GROUPS OF p-GROUPS

THE FITTING SUBGROUP 0F AUT C AND HOL G
QUATERNIONS AS CLOSED STABILITY GROUPS
APPENDIX

BIBLIOGRAPHY

iii

Page

20
39
46
60

69

I.

II.

Relations:

5

in

Operations:

[x .y]
[X ’01]

[H,A]

n
[H,‘A,...,A‘]

[HzK]
H/K
H\K
H x K
in

I‘ll

INDEX OF NOTATION

Is a subgroup of

Is a proper subgroup of

Is a normal subgroup of

Is isomorphic to

Is isomorphically contained in

Is an element of

The subgroup generated by
The image of K under the mapping f.

x Sx

Subgroup generated by all [h,a], h E H, a E A.
n-l

[[HiA,...,Al,A]

The index of K in H.

The factor group, H mod K.

The elements of H not in K.
The direct product of H and K.
The order of the element h.

The number of elements in H.

iv

III.

Groups:
Aut G
CH(K)
CAut C(H/K)
Fitt G
G'
Hol G
Hom(H,K)
Inn G
MG)
00‘)
Symm

Z(G)

The

The

The

The

The

The

The

set of prime divisors of ‘H\.
exponent of H.

inner automorphism induced by h.
set of all “h’ h 6 H.

mapping h a h for all h E H.
mapping sending everything to l.

mapping a restricted to H.

Product, not necessarily direct product.

Direct sum

n

The

The

divides m

automorphism group of G.

centralizer in H of K.

[a E Aut G‘h-lha E K for all h E H].

The

The

The

The

.The

The

The

The

The

Fitting subgroup of G.

commutator group, [G,G].

Holomorph of G.

set of homomorphisms from H into K.
inner automorphisms of G.

Frattini subgroup of G.

cyclic group of order n.

symmetric group on n symbols.

center of G.

INTRODUCTION

This dissertation arose from an effort to characterize the
stability groups of finite groups. In [10], Kaloujnine defined
stability groups for normal chains and showed that they are nil-
potent. Later, in [S], Hall defined stability groups for arbitrary
chains and showed that they too are nilpotent. Thus the question,
of which nilpotent subgroups of the automorphism group of a group
can be stability groups, arises naturally. In [6], Hall and
Hartley investigated what groups may be subgroups of stability groups
of infinite length chains.

In Chapter I, we define the stability group of a chain 3
and state some of its elementary properties. If A is a subgroup
of a stability group, we define A, the closure of A, and develop
some properties of those automorphism groups for which AI= A.
Using a method of'Schmid [14], we give some examples of stability
groups. As a partial answer to the question, "What nilpotent
groups may be stability groups?", we see that any abelian group may
be a closed stability group.

Stability groups of p-groups are also p-groups. Hence,
Chapter II deals with p-groups of automorphisms of a p-group that
are stability groups. The beginning of the chapter contains a
condition under which the centralizer in Aut G of a normal sub-

group H of G is a stability group and two instances when the

group of central automorphisms is a stability group. In [11],
Kaloujnine characterized the stability groups of elementary abelian
p-groups as those automorphism groups A such that

(i) A is a p-group.

(ii) A is normal in every p-Sylow of Aut G that contains

A.
(iii) A is the intersection of all p-Sylows of Aut G
that contain A.

We refer to those automorphism groups of p-groups which satisfy i,
ii, and iii as K-type automorphism groups. A proof of Kaloujnine's
characterization is given and it is shown that for a p-group those
automorphism groups of K-type are stability groups. In general a
p-group has stability groups that are not of K-type because if G
is a p-group, all of whose stability groups are of K-type, then G
is elementary abelian or cyclic of order p2.

In Chapter III, we consider the Fitting subgroup of H01 G
and of Aut G. Fitt(Hol G) is shown to be the product of Fitt G
and the product of all stability groups of characteristic chains.
Fitt(Aut G) is characterized when G is a p-group and we determine
two conditions under which Fitt(Aut G) is a closed stability group.

Chapter IV begins with two examples, one of which is a 2-
group G and a normal chain 3 such that Stab(s) is isomorphic
to the quaternion group of order eight. The bulk of Chapter IV
consists of a proof that the quaternion group of order eight is not
a closed stability group for a normal chain in a 2-group.

Several examples are listed in the Appendix.

CHAPTER I

DEFINITIONS AND ELEMENTARY PROPERTIES
OF STABILITY GROUPS AND CLOSED STABILITY GROUPS

In this chapter we introduce some elementary properties which
will be used in later chapters. All groups considered are assumed to
be finite.

Definition 1.1. Let s: G = Go 2 G1 2...2 Gn = 1 be a chain
of subgroups for an arbitrary group G. We define the stability group
of 3, written Stab(s) by:

)a g.G

Stab(s) = {a 6 Aut G|(giG1+1 = 1 f+1

for all gi E Gi’

The process of finding stability groups may be simplified by:
Theorem 1.2. Let A = Stab(s) with s as above, and

9 E Aut G. If s9 signifies the chain

9, _e e 9-
s .G-G02G12...2Gn-l

then

A6 = Stab(se)

Proof: Let a t Stab(s), B = Stab(se), and g1 6 G1,

-1
9 6 9 o9 _ 09
(81G 1+1) ' (giG 1+1)

_ e
’ (giGi-l-l)

-1
3+1)9 0’9 = geGe and A9 s Stab(se) = B.

The same argument gives that

1
) = Stab(s)

-1 -
B9 9 6

II
>

s Stab(s

Operating on the containment by 9 gives B S.Ae, and A9 = B = Stab(se).

Using 1.2, we are able to simplify the process of finding all
the stability groups of a group G. Any series is a subseries of at
least one nonrefinable chain in G. Aut G induces a permutation
group on the chains of G. By 1.2, the stability groups of any two
chains in the same permutation orbit are conjugate in Aut G. We
first pick a chain from each orbit of nonrefinable chains. Then, we
find the stability groups of all their subchains. Conjugation of
those stability groups gives all possible stability groups, because

any chain 3 is a subseries of some nonrefinable chain 3 We may

1.
permute s1 to one of the nonrefinable chains 82 for which we have

computed the stability groups of its subchains, i.e. sa is a sub-

chain of 32 for some a 6 Ant G so we know Stab(sa). By 1.2,
-1
Stab(s) = (Stab(sa))a is one of the Specified conjugates.

Definition 1.3. Let s be the series 5: G = G0 2 G1 2...2

S(s) = {a E Aut G‘G: = Gi for all i = 0,1,...,n}

Theorem 1.4. S(s) S‘N t G(Stab(s)).

Au

6

Proof: Let A = Stab(s) and B E §(s). By 1.2, A = Stab(sB).

Se = 3 since 8 fixes each subgroup of the chain. Thus A3 = Stab(s)
and B E NAut C(A>°
2
In Example 1 of the Appendix with s: D 2 <3: ,y> 2 <x2> 2 1

4
we see that S(s) g “Aut G(Stab(s)). Thus we do not in general have

equality in 1.4.

Corollarygl.5. If each Gi is characteristic in G then
Stab(s) <1 Aut G.

Proof; If each Gi is characteristic, then Aut G = 9(3).

Thus, by 1.4 Aut G SN (Stab(s)), and Aut G normalizes Stab(s).

Aut G
Definition 1.6. Let A s Aut G. Set

yGA0=G , and

v G A1+1 = [y C A1, A] for i 2 0 .
For a group H, n(H) will denote the prime divisors of H.
Theorem 1.7. Using the notation of 1.1 and A s Stab(s),
we see:
(i) ‘YGAiCGi , i=0,1,...,n
(ii) [G, A] c Fitt G
(iii) The prime divisors of A are the same as the prime

divisors of [G, A], i.e. fi(A) = "([G, A]).

nggf: See Schmid [13].

Definition 1.8. Let A s Aut G. A stabilizes a series 5
if A s Stab(s). Following Schmid [13], A E It if and only if A
stabilizes a chain. Hence I denotes the set of all subgroups of

G
the stability groups of G.

 

Definition 1.9. For A 6 It, define a by g = {Y G A1]:::)
where n(A) is the first integer SUCh that y G An<A> = 1. Define

A; the closure of A, by

X'= Stab(s)

We say a stability group A is closed if A = A)
We record the following properties of closure in:

Lemm§:1.10. Let A E 36. Then

(i) A 4 A

(ii) If .A S B 5.x, then y GBi = v G A1 for all i.

(iii) A = A

(iv) If a e N (A) then (y 6 Ai)3 = y c Ai.

Aut G
(v) If AdAutG then KaAut 6.

(vi) A and A. have the same prime divisors.

(vii) §(s) = N

xo -1 xo

ll
3‘
X

x [x, a] E x v G AI+ and

[G, A,...,A]a = [6“, A“,...,A“]

.+ '+
(x v G A1 1)a = x [x, a] v G A1

ll
X
.4
C)
:5
.3:

Thus a E Stab(s).

(ii) Since B s A'= Stab(s), 1.7 says v G B1 is contained

in the igh term of the chain which is v G A1.

Induct on i for the other inclusion.

v G B0 = G = v G A0 .

A generator of v G A1, i >'0, is of the form [x, o] with

x E v G A1.1 and a E A. By induction x E v G Bi.1 and since

i-l, B] = v G B1. By induction v G A1 =

atAsB, [x,a]€[yGB
V G B1 for i 2 0, and consequently A.= Stab(y G A1) =

Stab(y 0 Bi) = E.

B and i=1? by (ii).

> II
II

(iii) Let B =X. Then

O

(iv) Induct on i. v G A

(yGA0)B=GB=G=yGAO.

G so for B E Aut G,

For i >,o’ take a generator [x, a] of v G A1 with

x E Y G Ai-l, a E A. [x, a]8 = [xB, as] and by induction

8

x E v G A1-1. Since 3 normalizes A, we have

1

(v G A1)8 5 [v G A1- , A] = v G A1 as required.

n(A)
i=1 '

part iv says (v G A1)B = v G A1. We now have

(v) Let a e Aut o. By 1.2, (A)6 = Stab((y c Ai)B)

Since 3 6 NAut G(A),

(A)8 = Stab(y c A528? = A .

Hence A <) Aut G.

(vi) By ii, [0, A]

[G, A]. Since A, A'E Tb, 1.7 gives

n(A) = n([G, A]) and n(A) n([G, A]). Thus n(A) = n(A).

(Vii) BY 1", N (X) S§(§). The opposite inclusion is

Aut G
1.4.

We record the following important theorems for future re-

ference.

Theorem 1.11. (P. Hall [5]). If s is defined by

 

s: G = Go 2 G1 2...2 Gn = 1 then Stab(s) is nilpotent of class
less than or equal to (2).
Theorem 1.12. (Kaloujnine [10]). If each Gi'<lG and
Gn = 1 then the class of Stab(s) is less than or equal to n-l.
The following lemma and method were introduced by Schmid in
[14].

Lemma 1.13. Let H and K be normal subgroups of G. If

 

K s H and c =CG(H/K) then c (H/K) so (G/C).

Aut G Aut G
Schmid notes that if H < G, CG(H) s H and L 2 H then

CAut C(L) = Stab(G 2 L 2 1). To see this let K = 1 in the above
learns. Then CAut G(H) ‘CAut G(G/H). Since L 2 H,

CAut C(L) ‘ CAut Gm) ‘ CAut G(G/H) S CAut: C(G/L)
Thus CAut G(L) = Stab(G 2 L 2 1) as stated.

If G is solvable,by Gorenstein [9, 6.1.3], CG(Fitt G) s
Fitt G. Thus if H = Fitt G and ‘L 2 H = Fitt G , the above method

gives that C (L) = Stab(G 2'L 2 l).

Aut G
For G a p-group, Thompson has proved (see Gorenstein [9,
5.3.11]) the existence of a characteristic subgroup C, called the
critical subgroup, such that
(1) class of C S 2 and C/Z(C) is elementary abelian
(ii) [6, C] s Z(C)
(iii) CG(C) s Z(C).

With L = C in Schmid's method, C (C) = Stab(G 2.C 2 1).

Aut C

When G is supersolvable with a maximal abelian normal sub-

group M, CG(M) = M. If not, CG(M) 2 M, because CG(M) 2M. We

could refine the normal series G 2.CG(M) 2 M 2 l to a chief series.
Then there would be a normal subgroup N with CG(M) 2‘N ; M.
Supersolvability forces the chief factor N/M to have order a prime
p. Thus N = <M, x>' where x E CG(M) and xM generates N/M.
Since x commutes with M, we get G D'N 2 M and N is abelian, a
contradiction to the choice of M.

Thus if G is supersolvable and H is a maximal abelian

normal subgroup CG(H) = H. Schmid's method says that

Definition 1.14. For H 2 K,

1

C t C(H\K) = {a 6 Aut G‘Ha = H, Kg = K, h- ha e K for all h E H]

Au

The next theorem gives us our first example of a closed
stability group.

Theorem 1.15. Let H $.§(G), the Frattini subgroup of G.
If B = CAut G(G\H) then B is a closed stability group.

Proof: In [14], Schmid has shown that if A = cAut G(c/i)
then V G An = 1 for some n. Since [G, B] S‘H s Q, we see that

n -

B 5A and thus v c B — 1. B s B ‘CAnt G((:/[(:, 8]). Since
(mu) =B and B =1?

[G, B] sH, we have c (G/[G, B]) gC

Aut G
is a closed stability group.

Aut G

In 1.15 B may be the identity subgroupiwhich is always a
closed stability group for the trivial series G 2 1.
we will see in 1.21 that the question "Which nilpotent groups

may be closed stability groups?" may be reduced to "Which p-groups

10

may be closed stability groups?" First we must develop a few lemmas.

Lemma 1.16. Let A 6 16 and n be the prime divisors of

 

A. If K is an A-admissible n'-subgroup of G, then A fixes K
pointwise.
i i , n
Proof: v K.A S v G A for all 1, and v G A = 1 for

some n, so AlK E TK' By 1.7, n(AlK) = n([K, AlK]). [K, AAK] is

contained in the n' —group K, and n(A‘K) S n(A) = n. Thus A‘K = 1K,

since it is both a n and n' group.
Lemma 1.17. If' A E It then y C A1 = y'G(An)i x yG(Afl.)1
for i = 1,2,...,n. Here n and n' are any two disjoint sets of

primes with n U n' = n(A) and AT1 = n S , A“, = H S with
pEn p qtn'
St the t-Sylow subgroup of A.

Proof: Induct on i. [G, A] 2 [G, An]-[G, An'] since

Afl,.Afl, $.A. Theorem 1.7 says n = n([G, A]) and n([G, Ah'J) = n'.

The two normal subgroups [G, An] and [G, An'] must now have trivial

intersection so the subgroup generated by the two is a direct product.

8

A generator of [G, A] must have the form g-1ga with g 6 G,

a E An and B t An' as A is nilpotent and thus the direct product

of’ A and A ..
1'1 Tl

8-1808 = g-lgo’(gm).1(gw)B e [G, A“) . [G, A".] -

Hence [G, A] = [G, An] X [G, An,].

1

For the case i+l 2 2 we again have y'G A1+ 2 yG(An_)1+1 X

1+1 i+1
w G(Afi,) , because A 2 A11 x An“ Since y G(An) s [G, An]

i+1

and y G(Afl.) S [G, An'] we again have that the subgroup generated

by the two is a direct product.

11

Take an arbitrary element (x,y) E y G(A“)i X y G(Ah.)i ,
a E A“, and a 6 Afi.. Since [G, A".] is a n' group that is ATT

admissible, we use 1.16 to see (xy)-1(xy)a8 = (xy)-1(x°[y)B

1x -1 a )8,

(x y x“ e [G, An]°’ = [c°’, 11:3 = [G, A 113 which by 1.16 is

fixed pointwise by B so (xy)']'(xy)a'B = y 1x 1xaya. Since

X1 xa E [G, A n] and y 16 [G, Afl.] they must centralize each other

- '+ H-
and (xy)1(xy)°’B = x 1x “y 1y3 t yG(An)1 1 - y G(Afl.) 1. Thus

i+l i+ +1 . . .
y G(A") X y G(An.)1+ = v G A and by induction the theorem is
proved.

Definition 1.18. If p is a prime then Op(G) or just Op
is the largest normal p-subgroup of G. If x is a set of primes

then On(G) is the product of O p(G) for every p é a.

Theorem 1.19. If A is a closed stability group with A1"

n A 11)
and An , as in 1.17 then A= Stab(y G(ATT ) 11:) (0 and
n(A .) “
A . =Stab(y G(Afl,)11=)0" .
" , nun)
Proof: We know from 1.10. i that Afl s Stab(y G(A n) )i-

i n(An) ~10
If a t B = Stab(y G(An) )1=0 then 3 g“ t [G, An]. By 1.16

a fixes the characteristic subgroup On'(G) pointwise and the normal
nilpotent n'-group [G, A“.] is contained in On'(G)° Let

(x,y) E y c A1 = w G(An)1 x y G(An.)i, i 2 1.

y-lx- lxayo

(XV-1) (X300

-1 -l a
YXXY
-1 a

X

X

because x-1xa't [G, An], y c [G, An'] and the two groups centralize
each other. Thus B S Stab(y G A1) =A1T X A",. Since n(B) 8

n([c, B3) c n[c, An] = n, B SA". We now have that AT1 = Stab y(G Ari!)

12

A i
and since 11 was an arbitrary set of primes, An' = Stab(y G(An') ).

Theorem 1.20. If A and B are closed stability groups

 

with n(A) and n(B) disjoint, then A and B commute and
<A,B> = A X B is a closed stability group.

nggfz In [13], Schmid proves that C = <A,B> = A X B
stabilizes a series and thus v G Cn = l for some n. Lemma 1.17
says yGCi=vGAiXyGBi forall lsisn. Let
(T = Stab(y G Ci)“$).

n(C) = n([G, C]).

By 1.7, n(A) = n([c, A]), 1103) = n([c, B3) and

We calculate the prime divisors of C by:

ME) n(C)

= "([G, C])
= n([G, A] X [G, B])
= "([G, A]) U MIG. 3])

= n(A) U n(B).

C stabilizes a series and is therefore nilpotent. We have

C = Cn(A) X Cn(B) where Cn(A)(Cn(B)) is the n(A)(n(B))-Hall sub-

— - 1
group of C, For a 6 Cn(A) and i 2 O, [y C A , ((1)15

.+ _
GC11 because 'yGAi'iwyGCi and C

[Y G A “W

SY

EMA) 1
in(C) i

stabilizes (v G C +)1 . By 1.7 and 1.17, [v G A , <a>] s

(v GA“-1 X y G 8H1) n 0 n(A)(G) which is v G Ai+1. Thus

5 sStb A “m- . " — '
n(A) a (yG )1=A Since G 2A and Cn(A) 18 the

n(A)-Hall subgroup of c, E = A. Likewise c = B, and

n(A) T1(B)

A XB =E= Stab(y cc )"(C).

Corollary 1.21. A is a closed stability group if and only

if for every p E n(A) a p-Sylow of A is a closed stability group.

13

"a" This direction is 1.19 with n = {p}.
"=" By 1.20, A is a closed stability group.

Theorem 1.22. Let G be the semidirect product of H <IG

 

and K, A = Stab(H 2 H1 2...2 Hn = 1), and

B = Stab(G = HK 2 G1 2...2 G8 = H 2 H 2...2 Hn = 1). If the auto-

1
morphisms of H induced by K are contained in CAut H(A)’ then
B is the semidirect product of Stab(G 2 G1 2...2 GS = H 2 l) <JB

by {¢a= kh e khalo t A] e-A.

First note the following lemma:

Lemma 1.23. With the notation and conditions of 1.22, @a
is an automorphism of G = HK.

Prggf: $0 is well defined since every element of G is a
unique product of the form kh with k e K and h E H.

$0 is a homormophism. Let k h and k h be elements of

k l l 2 2
_ 2
K H. klhlkZhZ — k1k2h1 h2 so
c kg
a _ o
2 a o
' k1k2(h1 ) h2
ak
_ 2 o
' k1k2h1 hz

because A commutes with automorphisms of H that are induced by

O 0 ¢
CY = (1’ at = a a

$0 is one-to-one and since G is finite ¢o is an auto-

B
morphism. If 1 = (klhl) “ = k h“ then h“

-1
1 1 1 - k1 E H n K — 1 and

klhl = 1.
Proof of 1.22: Let , A. 3' c = ’ n a n
o B E In e $08 ma¢8 a ¢o
is a homomorphism from A into Aut G. If 1.G = $0 then

1 = = . n A u - . .
H (¢a)‘H a Thus a ¢a is an isomorphism, i e

14

=‘ t A .
A {Bald 1
Let y E B and y)“ = do Since Y Q B, a E A. ¢ ‘1 fixes
a
K pointwise so it centralizes G/H. ¢ _1 was defined to have the
a

action of 0-1 on H so ¢ _1 E Stab(G 2 H 2 H1 2...2 Hn = 1) s B.
a
y¢ _1 is an element of B that centralizes H so

0’
w 1 e Stab(G 2 c1 2...2 H 2 1) and y = (m 1M has the required
' r o

a a
form. All that remains to Show is that {¢a\a €.A] fl
Stab(G 2 G1 2...2 H 2 1) is trivial. For $0 in the intersection
= d = .
(¢Q)AH 1H an ¢(1H) 13
Theorem 1.24. If C = Stab(H X K 2 H X K1 2...2 H X Ks =

0’

(ha,k) for every

$0
H 2 H1 2...2 H 1), A = [¢a\(h,k)

n

o e Stab(H 2 H 2...2 H = 1)} and B

1 n

every 6 t Stab(K 2 K1 2.. .2 KS = 1)], then C is the semidirect

product of Stab(H X K 2 H 2 1) <1C by A X B.

{13l(h»k>* = <h.k3) for

Proof: By 1.22, C Stab(H X K 2 H X K1 2...2 H 2 1) - A.

Let D = Stab(H X K 2 H X K

1 2---2 H X Ks = H 2 1), y E D and

y the automorphism induced by y_1 on HXK/H" K. The auto-

morphism )_tI fixes H pointwise and induces y-l on HXK/H

Y
so Y_:T E D. Examine the action of y v-TI. on the cosets kH.
Y Y
k E K.
111—1 Y Ti
um) Y =(kin V =kH.

YW-II centralizes KXH/H and H, so yv-ji E Stab(H X K 2 H 2 1)
Y Y
and Y‘Y‘I'Z—f ¢;€Stab(H XK2 H2 1): B. Thus
Y
c =Stab(HXK2 H2 1) -B-A.

B'A = B X A since ¢awe = ¢B¢ and A n B = 1.
a

15

Let ¢B¢a E B-A n Stab(H X K 2 H 2 1). Then for every h E H,

h = hv9¢a = h¢ ¢B¢1 = khk

B

a = ha and a = 1H. For every k E K, k = k

for some hk E H, since ¢B¢a E Stab(H X K 2 H 2 1). B E Aut K so
k3 = k and a = 1x' Thus v8¢a = 1 and c is the appropriate
semidirect product.

The next theorem could possibly be used to build up arbitrary

stability groups rather than just closed stability groups.

Theorem 1.25. Let G be nilpotent and B the isomorphism

 

Aut G 1 ll Aut(S ) where Sp is the p-Sylow of G. A SAut G

P
MM
is a stability group if and only if AI = H Stab(sp) where sp

MM
is a series from the p-Sylow to the identity.

Proof: Since the theorem is trivial for p-groups, we induct

on the number of primes dividing \G‘. Let s: G = G 2 G 2...2 G = 1

0 1 n
be the series and Gi = Hi X Ki where H1 is the p-Sylow of Gi
and K1 the p'-Ha11 subgroup of Gi' We use the isomorphism

X Aut K , h E H

Aut G‘” Aut HO X Aut K0. For (61,62) E Aut H0 0 i i

and k. E K,
1 1

_1 (91,62)
(91.92) E Stab(s) e (hi’ki) (hi’ki) E Hi+1 x KM1
9 9
-1 1 -1 2
° (hi hi ’ ki ki ) E Hi+1 X Ki+1
e 91 6 Stab(H0 2 H1 2...2 Hn = 1).
92 E Stab(K0 2...2 Kn = 1)
By induction Stab(KO 2...2 K = l) = H Stab(s ). Thus
0 q
ellKol
(81.92) E Stab(s) a (91,92) 6 Stab(sp) X q‘i‘IIG\Stab(sq).

qr?

16

The following theorem is very useful for producing auto-
morphisms and stability groups.
Theorem 1.26. Let A s Z(G) the center of G, g = gA and

a E Stab(s) then

Stab(G 2 A 2 1)=~ Hom(G/A, A)

by the mapping a « fa: g « g-1ga.

Proof: (i) fa is well-defined. Let g = EZ'E G/A.

ra(§3) = (ga)-1(ga)a = a-lg-lgaa, since a fixes A pointwise.

Because a E Z(G), fa(g;) = 3-130 = fa(é).
(11) f is a homomorphism. Let gl,é2 E G/A.
a
f (t s ) = (g g )'1<s g )a
a 1 2 1 2 l 2
_ -1 -1 a a
_ g2 81 3132

-1 a -1 a -1 a i
g1 glg2 g2 since g1 g1 E Z(G).

Thus fa(s1 82) = fa(sl)fa(gz).
(iii) The mapping a a fa is a homomorphism. Theorem 1.11
says Stab(G 2.A 2 l) is abelian. For a,B E Stab(G Z.A 2 1) and

i 6 CIA,

- _ -l as
fae(g) g B
s'lga(sa).lsOla
-l)agBa

1

- a
g g (a

-1 -1

B Ba(s BB)“
-1 -1

g Bag BB

because g-lgB E A, which is fixed pointwise by 0- Hence

l7

faa(g) fa(g)fe(g)

(faf5)(g)

(iv) o « fa is a one-to-one mapping. When we use a multi-
plicative notation for G/A, A, and Hom(G/A, A), the identity of
Hom(G/A, A) is cl, the constant map to 1. If fo = c1 then
1 = fa(g) = 3-130 for every g E G. Thus g = g01 for every g E G,
and ker(a a fa) is the identity of Aut G.

(v) a -+ fa is onto Hom(G/A, A). For B E Hom(G/A, A),
define I: G etc by

By = BBB

B

for every g E G, where a is the composition of G etc/A a A.

I _ B S
(8182) - 81828182

= B 3 - 1 v
g1g13232 ' 8182

because g? E.A s Z(G). To show that Y is an automorphism, take
x E G such that l = x‘9 = xxa. This implies x-1 = x8 and is in
the image of a. The image of B is in A and thus x.1 is in
the kernel of 5- Since x.1 E ker B, 1 = xx8 = x-l = x. Thus ‘I
is a monomorphism and since G is finite, p is an automorphism.
For g E G, g-1g¢ = g-lggB = g6 E A. If g = a E A, the equation
becomes a-laI = a8 = l and aI = a. These calculations Show

1 EStab(G 2A 2 1).

f (:2) g'lgl

8.1888 = (B)E.

so I maps onto g as required.

18

It might be noted that Hom(G/A, A) = Hom(G/G'A, A) in 1.26
because A is abelian and thus any homomorfii ism from G/A into A
will have G' in its kernel.

Corollary 1.27. Let A s Z(G) with A = 69 z; <yi> where
m i=1
each <yi> is a cyclic group, and G/G'A = (+3 2 ocj> where each
_ i=1
<xj> is a cyclic group. Then

Stab{c 2A 2 1) e: («92: I: a((\§.\. \y.\))-
11 J 1

Here (\le, lyil) is the greatest common divisor of the

orders ‘52“ , and \y1\ , and q(n) denotes a cyclic group of order 11.
11921:: By 1.26, Stab(G 2 A 2 1} er Hom(G/G'A, A),

Hom(G) Z o’cj>, @2q1>) = (+3 2 2 Hom(éjz <yi>) since

Hom(H 29 K, L) 9" Hom(H, L) (+3 Hom(K, L) and Hom(S, T 6-) U) 2'

Hom(S,T) @Hom(S,U). Let 3:?“ = n, \y1\ = m and k = (n,m). A

homomorphism f E Hom(e-cj>, <yi>) is determined by its action on

I.

the generator 123- xj = y1 is a homomorphism if and only if

n
(Y?) = y?“ = 1. This in turn is equivalent to m‘nx. Since
(m,n) = k there are integers a,b such that am + bn = k. Thus

am), + bu), = k)U and Since min)" we have min), is equivalent to
k
mlkx or (y?) = 1. Since there is only one such set of elements ,

namely qm/g, we have: f E Hom(éj>, <yi>) if and only if

f(xj) E <ytyk>. If f(xj)= ym/k we see that fs(x

J
sm/k "
= y . f has order m and generates Hom(dj>. <yi>) -

) = f(xj) f(xj)

Lemma 1.28. Let A be an abelian p-group and X = <x>
a cyclic group of order pk > EXp A. If G = A X X then
B = Stab(G 2 A X {1] 2 1) is a closed stability group isomorphic

to A.

19

n
Proof: Let A = G) 2 <y1> where each <y1> is a cyclic
181
group and B = Stab(G 2 A X {1] z 1). By 1.27, B an): aqx‘, lyil).
1

Since \X\ > up A 2 A371" and both \x‘ and Ayi‘ are powers of p
1\) =- A.
In 1.26 one notices that [G, B] = <Image of f‘f E Hom(G/A, A)>.

(|x\, \y1\) = \y13. Thus Stab(G 2A x {1} 2 1) =69): o(\y
1

Since \Xl > \yil then [G, B] = A X {1] and B must be a closed
stability group.

Theorem 1.29. Any abelian group A is a closed stability
group for some abelian group G.

I_’_r_9_9_f: Let Ap be a p-Sylow of the abelian group A. By
the previous lemma, AI) is a closed stability group for a p-group
GP. Theorem 1.25 gives that [1 AI) is a stability group for
G - nGp, so y G(Ap)n = l for some n. Let D = Stab(y G A113) for

i = 0,1,...,n. By 1.7,
n(D) = n(lG. D1) ‘3 n(CG. AP]) = MAP) = {P} -

Since the q-Sylows of an abelian group are characteristic, 1.16 says

that both D and AI) fix all q-Sylows pointwise for q 9‘ p. Thus
i

y G(Ap) 3 Y GPCAP)i and V G D1 = Y chi for all i 2 1. Since

D - Stab(yGAi),
i i i 1
CD = GD = GA = G A .
V p I V (p) Y 9(9)

Thus the restriction of D to GP, DAG is contained in

Ap 8 Stab(y GP(AP)1). If d E D fixesp Gp pointwise, d = 1 be-
cause D fixes the other Sylows pointwise. By definition, D is
the closure of Ap in Aut G so D 2 Ap and we have Ap 2 DAG °‘
D 2 AP. Since every Sylow of A is now a closed stability group

1.21 says A is a closed stability group.

CHAPTER II

STABILITY GROUPS OF p-GROUPS

This chapter deals with stability groups of p-groups, which
we have seen must alSo be p—groups (1.7). By applying 1.25, we will
know all possible stability groups for a nilpotent group G provided
we know the stability groups for each p-Sylow subgroup of G.

Theorem 2.1. Let G be a p-group, p a prime. If P is
a p-Sylow of .Aut G then P is the stability group of a chief series.

2522;: The semidirect product GP is a p-group and there-
fore has a lower central series, GP = To 2 F1 2...2 Tn = 1 for some
integer n. By intersecting this with G we obtain
8:62F1flG2I‘20G2...2I‘nflG=1. Since FquP wehave
FinG<G for all i and s isanormal series. Since (F1)

is the lower central series of GP, Fi+1 2 [F GP] 2 [F1, P] for

i’
all 1. Because [3, P] SC for any g E G, we see that P
stabilizes s and consequently any refinement of s. In particular,
if S is a chief series that refines s, by 1.7, Stab(fi) is a
p-group containing P. Thus P = Stab(fi).
Corollary 2.2. Let G be a p-group of order p“, and P
a p-Sylow of Aut G. If c is the class of P then c s n-l.
2322;; The previous theorem says that P is the stability
group of a chief series which has length n, because lG‘ = p“.
By 1.12, c S n-l.

20

21

Aut G induces a permutation group on the chief series of

6 is defined in 1.2.

G by a: 8.4 s6 for every 9 E Aut G, where a
By 2.1, a p-Sylow, P, of the automorphism group of a p-group G is
the stability group of some chief series a. Let Ps be the set of
chief series in the permutation orbit that contains 3.

Theorem 2.3. Let G be a p-group. If k is the number of
p-Sylow subgroups of Aut G, then k is less than or equal to the
number of chief series in any Ps’ where P8 is defined above.

2522;; Let P be a p-Sylow of .Aut G and
s: G 2G1 2...2 Gn 8 1 be a chief series with P = Stab(s). Since
all p-Sylows of Aut G are conjugate and P9 = Stab(G 2 G: 2...2 G: = 1)
for 9 E Aut G, we see that each p-Sylow is the stability group for
a chief series in PS. Thus the number of p-Sylows of Aut G is
less than or equal to the number of elements (chief series) in the
orbit Ps'

The following definition was motivated by Kaloujnine's char-
acterization of the stability groups for an elementary abelian p-
group (See 2.20).

Definition 2.4. Let G be a p-group, and A s Aut G. A
is said to be of K-type if

(1) A is a p-group.

(2) A is normal in every p-Sylow of Aut G that contains A.

(3) A is the intersection of all p-Sylows of Aut G that

contain A.

Theorem 2.5. If G is a p-group and A is of K-type then

A is a closed stability group.

22

Proof: Since A s P for some p-Sylow of Aut G,

i
YGA gyGP1 forall i. By2.1,yGPn=1 forsome n and

n «—
thus v G A 1 = 1 for some n S n. A = Stab(y G A1) is a p-group

1

since n(X) - n([G, A]) I {p}. We now have A S A 5 Po where P0

is a p-Sylow of Aut G. Let B = <l’i\Pi 2 A and Pi is a p-Sylow

of Aut G, 0 s i s r>. Since A is normal in each Pi’ A 4B.

By 1.10.iv (v G A1)B = v G A1 for every 8 E B. According to 1.2,

A Stab(y 6 Ai)

Stab((y c A59)

= (106-
Since each P1 that contains A is a p—Sylow of B, there exists
a, _ _ a, a,
a E B such that P 8 P for each 1. Hence A = (A) s P 8 P .
i r 0 i 0 i
This forces A s (1 Pi = A. Thus A - A, which is equivalent to

i=0
saying that A is a closed stability group.

Corollary 2.6. If P is a p-Sylow of Aut G and G a
p-group, then P is a closed stability group.

1_’_1;<_>9_f_: P is trivially of K-type.

Corollary 2.7. Let \G\ 8 p“, and .A s.Aut G be of K-type
and of nilpotence class k. If the class of G is c then
c-l S k S n-l.

2592;: Since A is an intersection of p-Sylows of Aut G,
and every p-Sylow contains Inn G, A 2 Inn G. The lower bound comes
from the fact that Inn G has class c-1, and the upper bound from
the containment of A in a p-Sylow of Aut G which by 2.2 has
class bounded by n-l.

By intersecting all p-Sylows of Aut G, we are led to the

smallest possible K-type stability group, Op(Aut G) , which is

23

contained in every K-type stability group.

Corollary 2.8. If G is a p-group and A = 0p(Aut G),
then A, is a K-type stability group for a characteristic series.

2522;; OP(Aut G) = FKP‘P is a p-Sylow of Aut G] is normal
in Aut G. By 2.5, A = Stab(y G A1). Since A 4.Aut G, l.10.iv
gives (v G A1)6 = V G A1 for every 3 E Aut G, i = 0,1,...,n.
Thus each v G A1 is characteristic in G.

Let us note that if G ‘is a p-group and A is a p-group
contained in Aut G, then y G An = 1 for some n. If A is also
the set of automorphisms fixing pointwise some normal subgroup H,

1

then A centralizes H-y G Al/H-y G AI+ , i = 0,1,2,...,n. This

gives A s Stab(G 2 [G, A]°H 2 [G, A, A]-H 2...2 H 2 1) 513 t G(H) = A»

An
and A, is a stability group. In particular, if G is a p-group

and C (H) is p—group for some H 4 G, then C (H) is a

Aut G
stability group.

Aut G

1
Definition 2.9. If G is a p-group, 01(G) = <x E Glxp = 1>.

When G is understood 01(G) may be indicated by “i'
If G is an abelian p-group for an arbitrary prime p or
if G is a nonabelian p-group with p an odd prime then (See

Gorenstein [9, pages 178, 184]) CAut G(01(G)) is a p-group and

thus a stability group.
If G is a p-group and E is maximal with reapect to being
a normal abelian subgroup of exponent pn > 2, n = 1,2,..., then

(See Blackburn [3]) C is a p-group and thus a stability

Aut G(E)

group.

In [2], Adney and Yen investigated C (G/Z(G)) which

Aut G
they called Ac. There one may find the following definition and theorem.

24

Definition 2.10. G is purely nonabelian if it does not have
an abelian direct factor.

Theorem 2.11. For a purely nonabelian group the correspondence
0.2 fa (see 1.26) is a one-to-one map of Ac onto Hom(G, z),

Theorem 2.12. Let G be a p-group and Z be the center of

G. If G' 2 2 then
Ac -- Stab(G 2 z 2 1) e Hom(G/G', Z).

Prggf; G is purely nonabelian for otherwise its abelian
direct factor would be inside the center of G but not in the
commutator 6'. By 2.11, \Ac\ 3 \Hom(G, Z)‘ = ‘Hom(G/G', Z)‘.

Ac a CAM: G(c/Z) 2 Stab(G 2 z 2 1) =- Hom(G/Z, 2) by 1.26. Since
G' 2 z, Hom(G/Z, 2) .. Hom(G/G', 2). Thus Ac 2 Stab(G 2 z 2 1) =-
Hom(G/G', Z) and since the orders of both ends are equal,

Ac = Stab(G Z Z 2 1).

Corollary 2.13. Let G be a p-group, and Z be the center
of G. If ‘2‘ 8 p and \G\ > p, then AC = Stab(G 2 Z 2 l).

2522:: Since |G| > \Zl, G is nonabelian. Thus
1 #G' 4G and G' n 2 2 1. Since ‘2‘ = p, G' 22 and the pre-
vious theorem gives the desired result.

Notice that if G is cyclic of order p then Ac = Aut G
but Stab(G 2 Z 2 1) = Stab(G 2 1) = 1.

Now we will calculate the stability groups for cyclic p-
groups.

Let G =<x> be a cyclic group of order pt, where p is
an odd prime. It is known (See Scott [15, page 117]) that Aut G

t- +
is cyclic of order p 1(p-l) where a: x 4 x1 p generates the

25

p-Sylow of Aut G. The subgroups of G are
2

G '<X>2<Xp>20¢p

k (1+p)k
.a : x a x and any stability group is contained in <a>.

>.2,_,2 l. The reader should note that

Lemma 2.14. With the notation above, and 0 s i < 11+ k,

i an integer, i +'k = 1,2,...,t,

p1 pk-l
G! > _.
C<'a> ifik, <0 >
(XP >
Proof:
J <2": ~91 pl]L ozj -pi+pi(1+i>)j
0' “up jar” 0‘) =X
(X > .
1 j i+k
+ - -
=xp [(1 p) 11c<x13 >
. .+
a pi’l'_(l+p)J - l] E 0 mod p1 k
d (1+p)j - l E 0 mod pk
e (1+p)j a 1 mod pk
Since the multiplicative order of l+p in the integers
k k-l k-l
mod p is p , this last statement is equivalent to p divides
k-l

jor ajE<ap >.

Theorem 2.15. For an odd prime p, let G = <x> 'be a cyclic

group of order pt, and a: x a x1+p. If pk is the largest index
of two consecutive groups in a series a: G = GO 2 G1 2...2 G = 1,
k-l

t-k
then Stab(s) <0p > is cyclic of order p .

 

G . [G,:G, ]
+1
Proof: By the above lemma, C (-i-——) =<am(1)\m(i)- 1 1
<a> (; p
i+1
n-l 61
Thus Stab(s) = Fl C (————)
i=1 (a) G1+1
[Gi'ci+1]

<oan = maxf——————-——]>
-1 ' P

‘(ap >.

26

Let G = <x>' be cyclic of order 2t > 4. It is known (See

Scott [15, page 121]) that Aut G = <SI> X.<-I> where 51: x 4 x5,
- - J
-1: x.~ x 1. The reader should note that (51)]: x a x5 .

Lemma 2.16. With the above notation and i = 0,1,...,t-k,

21
(Gt>)= AutG if k=1

CAut G 2i+k 2k-2
r<x > (51) if k > 1
Proof: Case k = 1.

Since there is only one subgroup of any given order, all subgroups

are characteristic. Thus Aut G induces automorphisms on

 

i
2
ELIE—T C,(2), and A“ G CAut 0(2) = 1. Thus
<x > <X >
cAut G( 2L+1 )
i (X >
<x2
A = ————>
ut G CAut G 21+].
<X >
Case R > 1.
i
2 i i j i i j i j i+k
j .. <x >. -2 2 (SI) _ -2 2 S _ 2 (5 -1) 2
(51). CCAut c( zinc)"x 0“ ) ”x x "x ‘5“ >
<x >
e 21(5j - 1) e: 0 mod 21+k
.2 5‘1 - l a 0 mod 2k
@151 Z 1 mod 2k
Since the multiplicative order of 5 in the integers mod 2k
is 2k-2, this last statement is equivalent to 2k’2‘j or
j 2k-Z
(51) E<(SI) >
zk-Z <x21>
* = ._______
( ) Thus <(SI) >. CAut G( 2i+k ) n«<51> .

C! >

27
i

2
(X >
The following shows that -I é’cAut G( 2i+k ), k > 1.
(X ‘2
_2i 21 _I _2i+1 2i+k
x (x ) = x €i<x >
«21> «21>
Since <51) S CAut G W- and -I E CAut G 747- ,
<x > <x >
«21> «21> «21>
<5” ‘ CAut c—T-FT If k >- 2, CAut G W 5 0A,, 6—37) = <51>~
2 2 2
<x > <x > <3 >
«21> 2k-2
* . _______, =
From ( ) we have. CAut G( 21+k) <(51) >.
<}( ')

Theorem 2.17. Let G = <x> be a cyclic group of order

 

2t > 4. If 2k is the largest index of 2 consecutive groups in a

series 3: G = G 2 G 2...2 CD = 1 then

<SI> x<-I>=Aut G if k = 1

Stab(s) = k-2

 

 

<(5I)2 > if k> 1
Proof: By lemma 2.16,
n-l Gi
Stab(s) = D C ( )
i=0 A“ G Gi+1
G10
= CAut G(G, )
1 +1
0
Aut G if k = l
2k-2

_ k
where [Gio. Gi +1] — 2 ,

0

Of course when G is cyclic of order 2, Aut G = l and there

are no nontrivial stability groups. When G is cyclic of order 4

28

H

x.» x5" is the identity mapping. In this case
Aut G ' <-I> - Stab(<x> 2 62> 2 l).

The rest of this chapter deals with stability groups of
elementary abelian p-groups.

Kaloujnine, in [11], found a very nice characterization for
stability groups of elementary abelian p-groups. The proof given
here is mine since his was unavailable. By examples we see that
almost none of the interesting characteristics may be generalized
to stability groups of arbitrary p-groups.

In [17], Suprunenko also deals with nilpotent subgroups of
GL(n,p) which is the automorphism group of an elementary abelian
p-group of order p“.

Before proceeding to Kaloujnine's theorem it is necessary
to fix the notation to be used and to prove several technical facts.

Let G be elementary abelian and
s: G =Gn 2 Gn

2...2 G1 2,6 = 1 be a series of subSpaces. Con-

1 v1

2,..., 1(1)

G1. Extend this set to a basis of G2 where the new vectors added
2 2 2

are {v1,v2,...,vj(2)]. Continue this process so that

{vi,...,v:(i)] are the vectors used to extend the basis of G1

-1 0

struct a basis for G by picking a basis {vi,v ] for

-1

to a basis of' Gi'

Definition 2.18. For A S.Aut G, let
F(A) = {gec\g°'=g for all oeA3.

Lemma 2.19. Let G be elementary abelian with the above

notation. If A - Stab(s) then:

29

i i
(i) A 2 {T e Homzp(c,c)\va - vj e 614}
(ii) y c A1 = Gn-i

(iii) G1 = F(A).

Proof: (1) Every automorphism of Stab(s) must have the

required form. The matrix afforded such a T by the basis {vi}i j

2 2
under the ordering vim; . . . ”3(1) , v1 , . . . ,vj (2) , vi, . . . ”3101) has

the form .1 1) (2)

“k
3(2) >k

1(3 )K 01

J(n){ : * \

Since det T 8 1, T is an automorphism. T induces an automorphism
G

 

 

 

 

 

 

 

on each 51—41 which fixes the generators. T therefore centralizes
‘ i
Gi+l
each and T E A as required.
i .
(ii) A stabilizes s so yG A1 ‘Gn-i' We prove the
opposite containment by induction on i.
0 _ -
y G A - G Gn-O'
Let 0 9‘ w E G ..
n-i
Define T: vri'fil .. v’fifl + w
i i .
vj —. vj for all other i,3
w if s=n-i+1 and T=1

s s _
(VT)T'VT' EC

0 otherwise 8'1

30

By (1) T E A and w E [G A] which by induction equals

n-(i~1)’

[v G 111.1, A] =- V G A1 and we have Gn-i s y G A1 as required.

(iii) The last nontrivial term of a series a is always

fixed pointwise by Stab(s).

For u,v,2$usn,lsusj(p,). Define T: by:

v1 + v if i = p and j = v

(v?)Tu = j 1
J V i
J'

v otherwise

By (i) every T: E Stab(s). If 2 aiv1 is fixed by Stab(s), then:
11.ij
rafv1 =(za1vT)T”
1”ij i,jjJ V
i i l
=(2av,)+a”’v .
1.131 V1
1
Thus atv1 = 0, forcing a: = 0. Since p was an arbitrary integer
with 2 s u s n, the fixed point 2 agv; has nonzero coefficients
i,j
only when i = 1. Thus ' 2 aiv; E G1 and G1 is the set of fixed
i.j

points of A.

Theorem 2.20 (Kaloujnine [11]). Let G be an elementary
abelian p-group and A s Aut G. A is a stability group if and only
if A is of K-type.

23222: By 1.7.iii, A must be a p-group.

We next prove by induction on \G\ that A = Stab(s) <1P
for every p-Sylow P of Aut G that contains A.

If ‘Gl = p, then \Aut G] = p - 1. Since every stability

group of a p-group is a p-group the only possible stability group

is {1].

31

If P is a p-Sylow containing A, we may assume that P

fixes vi.

by A. P must fix a nontrivial vector because P is a stability

Since A $29, the points fixed by P are alBo fixed

group by 2.1. Remember that v: was an arbitrary nontrivial point

fixed by A, so we could just as well have chosen v1 as a vector

1
fixed by P.
Since <V}> is P and A-admissible, P and A induce auto-
* * 1

morphism groups P and A on G/<N1>u Let

* 1 1 1 ~ ~

B -= Stab(G/<v > 2 G /<v > 2...2 G /<v > 2 1) and indicate

1 n—l l 1 1
an image in G/<v:1l>. Since G °" G/<vi> G) 01>, any automorphism

*
in B may be extended to an automorphism of G in A. Con-

* * * * * *
sequently A maps onto B , i.e. A = B . By induction A = B <|P

* *
because A is normal in the p-Sylow containing P ., By lemma l.10.iv,

* *

Gi/<V:>, O s i s n is P admissible. Since A is a closed
A ~ * .
stability group 8(5) - NAut c(A ), 1.e. for t E P,
G /<v1> = (G /<V1>)w =(G¢ +i<vl>)/<v1>r This forces G = G* + <v1>
i l i 1 i l l ' i i 1 °

G1 and G: have the same dimension so G: = Gi and t E S(s).
By lemma 1.10.vi, S(s) -‘N (A) so P normalizes A.

Aut C

Now we show that A ==FKP|P 2 A, P a p-Sylow of Aut G].

Let P be the p-Sylow of Aut G of the form

P = {T E GL(G)‘matrix of T = I + E a, e,, for the basis
' . 1j 13
1>j
{v1 v1 v1 v2 v2 v3 vn ]
1, 2,000, j(l), 1,000, j(z)’ 1,000, j(n) 0.

Note that P 2 A. For 1 S i s n, 1 s j 5 j(i) let

32

i. i i
pj. v1 «‘vj
i i
Vj “V1
v: a v: (s,t) # (i,j) or (i,1)
We see that (¢;)-1 = I; and I} E S(s) = NAut G(A).
wi
i 3: i 1
“’1” ”NI? *1
' i
= ‘11ij
i i
= (V1 +'Gi—IH‘j
i
= vj +Gi-1 a
*1 i i
If B E fl.P j then a: v 41v, + G, 1, for all i,j. Thus
1,] .1 J 1‘
i i i i
. VJ ll
nPJsA. Since A=A st wehavethat A=n1>j.
isj isj

If we intersect more p-Sylows containing A then we still have
A = F){P\P is a p-Sylow containing A].

The characterization of stability groups of p-groups as
all those automorphism groups of K-type cannot be extended to abelian
p-groups. In Example 3 of the Appendix, Aut G is a 2-group for an
abelian 2-group G and thus the only stability group of K-type.
There is a closed stability group, namely «(na>, that is a proper
nonnormal subgroup of Aut G, which is its own 2-Sylow.

If G is an elementary abelian p-group, any series is
formed by adding a direct summand to the previous term in the series.
The following theorem deals with similar series for groups which

aren't necessarily even p-groups. For related results see Shoda

[16].

33

_T_h(eorem 2.21. Let G = H1 ®...® Hn with H arbitrary

i
groups, G1 = H1 Que) H1 for 1 s i s n, and

A =Stab{GzG.21]. If A=Stab{G=G 26 2...2G 21]
1. n U

i -l 1
then,
(1) Ai 4A and
(ii) A = AlAZ An_1 .
Proof: (1) Each G1 is A-admissible so
A A St b 2 G .
AsS(GzG121). By 1.4,AsS(Gzcizl)sNAutG( a(G 121))

(ii) To prove part (ii), induct on n , the length of the

series. For the case n -= 2, we have the trivial statement

A = Stab(G 2 G1 2 l) = A1. Let Q E A. By induction on n,
“Ga-1 20,1 an-Z where at E Stab(Gn”1 2 Ci 2 l) s Aut Gn-l' a
A i
= X : ... -9 ... .
Extend each a, to oi oi 1H“ (31. .gn_l.gn) ((31. .gn_1) .gn)
The following calculations Show that 221 E Stab(G 2 G1 2 l) for
i = l,...,n-2.
6:1 o1
(81,...,gi,1,...,1) = ((g1,...,gi,1,...,1) ,1) = (g1,...,gi,1,...,1).
Any g E G is of the form g = (x,gn) where x E 611-1 and
8n E Hn'
_1 a _1 _1 o. -1 a
s 8 i = (x is )(X 1.8 ) = (X x 1.1) E G. since
n n 1
oi E Stab(Gn_1 2 Ci 2 1).
(*) Let y = ”I ... an_2 6 A1 ... An_2. W‘s = a1 ... an_2 = 136 .
n-1 n-1

A —

The previous calculation shows that (t) 1* fixes Gnu1 pointwise.

A 5"].
Since each (11 E CAut G(G/Gi) s CAut G(G/Gn-l)’ 011 E CAut G(G/Gn ).

-l

34

I centralizes G/Gn-l’ therefore ($)-1¢ centralizes G/Gn-l'

(I)-1¢ must now be in Stab(G 2 Gn 2 l) and

-l
A A -1
t = ¢((y) I) E A1 ... An-2(An-l) as required.
Corollary 2.22. If G is elementary abelian and
A1 = Stab(G 2 Ci 2 1) then Stab(G 2 Gn_1 2...2 G1 2 l) = A1 ... An—l'
5
Examination of <x>, a cyclic group of order 3 , reveals

that Stab(s) need not be the product of stability groups of proper

subseries of s. For, if a: x 4 x4, by 2.15

32 32 33 2 3
Stab(<x>2<x >23<x >231)=<o>

2 2 3 2

3

Stab(<x> 2 <x3 > 2 l) = <03 >

3 3 2 2
Stab(<x> g <x3 > g l) = <03 > and

32 32

3
<0 > * <o > - <0 > .

Definition 2.23. A = Stab(s) is a minimal stability group

 

if A contains no other nontrivial Stability groups.

Since the stability group of a series contains the stability
group of a subseries one might expect any Stab(G 2 H 2 l) to be a
minimal stability group. In this direction, we have

Theorem 2.24. If G is an elementary abelian p-group and

 

H is a proper Subgroup, then Stab(G 2 H 2 1) is a minimal stability

group.

Proof: Let A Stab(G 2 H 2 1) and B be some other
stability group, 1 # B s A, which by 2.22 may be assumed to have the

form B = Stab(G 2 K 2 1). By 2.19.ii, H

[G, A3 and x = [G, B3.

Then B S.A implies K = [G, B] s [G, A]

H. 2.19.iii gives

F(A) = H and F(B) = K. Since B s A, K F(B) 2 F(A) = H and

K = H, i.e. B = Stab(G 2 H 2 l) A.

35

In Example 4 of the Appendix, Inn G = Stab(G 2 <a3> 2 l)
is not a minimal stability group.

There is an interesting relationship between the minimal
stability groups contained in the same p-Sylow of Aut G when G
is elementary abelian.

Lemma 2.25. Let G be an elementary abelian p-group,

A = Stab(G 2 H 2 1) and B = Stab(G 2 K 2 1). If A and B are
both in the same p-Sylow of Aut G then K 2 H or K S H.

2522;; <A,B>' is contained in some p-Sylow P of Aut G,
which by 2.20 must normalize A and B. Again by 2.20 every
stability group is closed and according to l.10.vii
§(s) = NAM G(Stab(s)). Thus A s NAut G(B) = S“; 2 x 2 1),
Assume that there exist 0 # vH E H\K and 0 # v: E K\H. Extend
{vi} to a basis of G by adding a basis of H and extending that

to a basis of G. Define T by T: v1 a'vl +'v

1 l H

v1 a v1 9 (iii) * (1,1)-

J J

i i i i l .
The computation ( 2 a V )T - z a v = a v shows that T central1zes
1.13 3 1,1 31 1 H
G/H. Also, if z aiv; E H then a: = 0 and T fixes H pointwise.
i,j
Thus by 2.19.1, T e Stab(G 2 H 2 1) = A. T E S(G 2 K 2 1) since
1 .

v1 4 v11+ vH E K. This contradicts .A S S(G 2 K 2 1), so either

K S'H or K 2 H.

In Example 3 of the Appendix, <TPB> = Stab(G 2 <x2y> 2 l)
and <TB> = Stab(G 2 <y> 2 l) are closed stability groups in the
same 2-Sylow of Aut G but <x2y> n <y> = 1.

Theorem 2.26. If G is elementary abelian, the product of
any two stability groups in the same p-Sylow of Aut G is again a

stability group.

36

Proof: Let A = Stab(G = GH 2 Gn 2...2 G 2 l) and

-1 1
B = Stab(G = Km 2 Km 1 2...2 K1 2 1) be contained in some p-Sylow
of Aut G. By Corollary 2.22, A = A1 An-l and B = B1 Bm_1
where A1 = Stab(G 2 Ci 2 l) and B]. = Stab(G 2 Kj 2 1). By 2.25

K32Gt or KsSGt forall S,t,OSsSm,OStSn. We insert

the sets Kj into the series (Gi)' Let (Hi) be the series after
I o . . 5

some of the K s have been inserted If H1 Kj and H1+1 i Kj ,

then by 2.25 H1+1 2 Kj. We insert this Kj, relabel the new series,

and continue the process until we obtain a series 3: (Hi) made up

of all the 6'8 and K's. By 2.22,

Stab(s) = Stab(G 2 H 2 1) Stab(G 2 HS 2 1) . Each

1

Stab(G 2 H 2 1) normalizes the others because they are all in the

1
same p-Sylow. We may, therefore, rearrange the product and repeat

the same group if necessary to get Stab(s) = A1...A B A-B.

“-1 .B1 0 O O m-l =

In 2.29 we will show that the characterization of stability
groups as those automorphism groups of K-type cannot be extended
from elementary abelian p-groups to arbitrary p-groups.

Lemma 2.27. Let G has p-group with HSG. If H con-

tains a nontrivial subgroup norml in G, then
Stab(GzH21)2l .

11mg: H is contained in a naximal subgroup M which is
therefore normal of index p. H n Z(G) 2 1 since H contains a sub-
group K 4 G which must intersect the center nontrivially. Take a
cyclic Subgroup <x> of order p from H n Z(G) . There exists a
homomorphism f from G onto <x> with kernel M. 1.26 says

that af: g -. g(g)f is an automorphism in Stab(G 2 <x> 2 1) , so

37

it centralizes the cosets of H. If g E H S M then

a

g f = g gf = 3.1 = g and “f fixes H pointwise. Thus

1 ¥ of E Stab(G 2 H 2 1).

Lemma 2.28. Let G be a p-group. G has a unique minimal

 

stability group if and only if G is cyclic.

Prggf: If G is a cyclic p-group 2.15 and 2.17 Show that
G has a unique minimal Stability group.

For the other implication let 1 # A be the unique minimal

stability group, and let M be a maximal subgroup of G, which is

1
nontrivial unless \G‘ = p and G is cyclic. By the previous lemma,

2 were some other maximal subgroup, then

1 #A sStab{G 2 M1 2 13 nStab(G 2 M

Stab(G 2 M12 1) 2 1. If M
2 2 1). This would say that A
fixes <M1,M2> = G pointwise, and would be a contradiction to

A # l.- Thus G contains only one maximal subgroup M, and M = Q
the Frattini subgroup. Because M is maximal, G = <M,x>. M = Q

is the set of nongenerators of G so <M,x> = <x>' as required.

Theorem 2.29. If G is a p-group for which every stability

 

group is of K-type, then G is elementary abelian or cyclic of
2
order p .

Proof: G is elementary abelian if and only if the Frattini

Q 1, so assume Q i 1. In a finite group G 2 Q So 2.27 gives

A

Stab(G 2 9 2 l) 2 1. Since the series is characteristic,
A14.Aut G. Since a normal p-group is contained in every vaylow,
and A is of K-type, A = FHP‘P is a p-Sylow of Aut G]. A must
be contained in any other nontrivial stability group since any other
is of K-type and thus an intersection of p-Sylows of’ Aut G. Thus

A is the unique minimal stability group for G, which by 2.28 is

38

cyclic. A check of 2.15 and 2.17 shows that the only K-type stability
groups of a cyclic group are the p-Sylows of Aut G and unless
\G‘ - p2, G has nontrivial stability groups that are not of K-type.

Along similar lines we have:

Theorem 2.30. If G is a p-group with Aut G containing
no nonidentity normal stability groups then G is elementary abelian.

2599f: Stab(G 2 Q 2 1) <1Aut G since 6 is a characteristic
subgroup. Unless Q = l, 2.27 says that Stab(G 2 Q 2 l) 2 1. Thus
Q = 1 and G must be elementary abelian.

If G is a p-group which isn't elementary abelian, 2.30
insures that Fitt(Aut G), the Fitting subgroup of Aut G, will be
nontrivial. Fitt(Aut G) will be investigated more extensively in
Chapter III.

Since K-type stability groups of a p-group G are inter-
sections of the p-Sylow subgroups of Aut G, a K-type stability group
must contain Op(Aut G). Even though a homocyclic p-group G is
very similar to elementary abelian p-group there is an example of
a stability group A containing Op(Aut C) but A is not of K-
type. I did find that for a homocyclic p-group G and A S.Aut G,

A is of K-type if and only if (i) A 2 Op(Aut G) and (ii) A is
a closed stability group. The proof has been omitted due to its

length.

CHAPTER III

THE FITTING SUBGROUP OF
AUT G AND HOL G

As was mentioned after 2.30 if G is a p-group, that is
not elementary abelian, then Fitt(Aut G) 2 1. This chapter is an
attempt to classify Fitt(Hol G) and Fitt(Aut C), using stability
groups of characteristic series.

Definition 3.1. The holomorph of G, written Hol G, is
the semidirect product of G and Aut G where a-lga = g“ is the
image of g under the automorphism a.

Definition 3.2. I: is the set of stability groups of char-
acteristic series of G.

Theorem 3.3. If A = A ETC Ai then Fitt(Hol G) = A-Fitt G.

i G

Prggf; In [14], Schmid has found that A is a closed
stability group for a characteristic series and that A 4 4 H01 G.
Fitt G 4 H01 G since Fitt G is characteristic in G. Because
both are nilpotent, we see that A'Fitt G S Fitt(Hol G), the sub-
group of H01 G generated by all nilpotent subnormal Subgroups.

Take ax E Fitt(Hol G), a E Aut G, x E G. The natural homo-
morphism from Hol G onto Aut G maps ax onto G. Since the

image of Fitt(Hol G) is a normal nilpotent Subgroup of Aut G,

a E Fitt(Aut G). Fitt(Hol G) stabilizes the series

a: Fitt G 2 [Fitt G, Fitt(Hol G)] 2...2 y Fitt G(Fitt(Hol G))1 2. .2 1
39

40

which is normal in Bel G and therefore characteristic in G.

3 ends in the identity because Fitt(Hol G) is nilpotent and
y(Fitt G)Fitt(Hol(G))n S y Fitt(Hol G)(Fitt(Hol G))n = 1 for some
n. Since [g,a-x] = g-lga x = g-1x-1gax = [g,a.flx], a flk

stabilizes s. [G, Fitt(Hol 6)] s Fitt(Hol G) FIG 5 Fitt C, so

a fix stabilizes the characteristic series

32: G 2 Fitt G 2...2 y Fitt G(Fitt(Hol G))i 2...2 1. Thus

a 11x 6 A s Fitt(Aut G) and because a 6 Fitt(Aut G),

fix 6 Fitt(Aut G) n Inn G <alnn G. Let H be the preimage of

Inn G F1Fitt(Aut G). H/Z(G) is nilpotent so x is an element of
H which is a normal nilpotent subgroup of G, i.e. x E Fitt G.
Since Fitt G s Fitt(Hol G) we have that a E Fitt(Hol G) FIAut C.
Now a stabilizes the characteristic series 32 so a E A as
required.

Corollary 3.4. If Fitt G is a n-group, Fitt(Hol G) is
a n-group.

{Egg£; By 3.3 Fitt(Hol G) = A-Fitt G. Thus Fitt(Hol G)
is the product of the n-group Fitt G and A which by 1.7 is also
a n-group. Fitt(Hol G) must therefore be a n-group.

The rest of the chapter deals with the question ”When is
Fitt(Aut G) a stability group?" It will first be necessary to
develop some technical lemmas.

Definition 3.5. If H s G let "h be the inner auto-

morphism of G induced by h and uh = {nh\h E H}.

Lemma 3.6. Let H s G, A s Aut G. Then [ ,A = .
---—- _1 ”n 3 “[H,A]
at 11110

1 3' -1 oz "(h°’)
Proof: The calculation x = (h- xa h)a = (h )axh = x

shows that a-lnha = n -
ho:

-1 ' _ g
[nhm] =- nh_la (11th nh_1nha - “11-th “[hfi] gives the required

conclusion.
Lemma 3.7. If A S.Aut G, then H i = [n i-l’ A].

Proof} By 3.6 [n .__, A] = n . = n i'
,ch1 1 [yGA ,A] VGA

Lemma 3.8. If A s.Fitt(Aut G) then A stabilizes a series
G 2...2 G = Z.
n
Proof: If A S Fitt(Aut G) then [B, A,..., A] = l for

every subgroup B s Aut G and sufficiently large n. Therefore
n

[Inn G, A ... A] = 1 for some large n. Repeated use of 3.7 gives
n n n-l
= ’ ‘J= m= ,‘Am=...=
1 [Inn G, A, A [36 [n[G,A] A n n

[c ,’A . . .A‘]
Thus V G An 5 Z(G) and A stabilizes the series

6 2 [G, A]Z 2...2 (y c A“)z = 2.

Lemma 3.9. If A s Fitt(Aut G) and n(A) r\n(Fitt G) = ¢
then A centralizes G/Z(G).

2592:; Let Z = Z(G). By 3.8, A stabilizes a series
G 2 G1 2...2 Z so A induces a subgroup A of
Stab(G/Z 2 cllz 2...2 2/2 = I). By 1.7 n(K) = n([c/z, K1) and
Fitt(G/Z) 2 [c/z, K] = [G,A]Z/Z. Thus [G,A]Z s Fitt c and
n(A) = n([G,A]Z/Z) §:n(Fitt G). Since A is an image of A, n(A)
is a subset of n(A) which is disjoint from n(Fitt G). A must
therefore be the identity and A must centralize G/Z.

Lemma 3.10. Let G be a nonabelian p—group, and Z be
the center of G. If B = Stab(G 2 Z 2 1) then [G,B] 2 01(2),
the elements of order p in the center of G.

Egggfg Let M be a maximal subgroup containing the proper

subgroup Z, the center of G. If 1 # z E 01(2): we see that there

42

is a homomorphism f from G onto i<z>’ with kernel M. By 1.26,
a: x ~»x xf is an automorphism and we see that a 6 B with
[G,a] - <z>. Thus 2 e [G,B] and (11(2) s [G,B].

Theorem 3.11. If G is a nonabelian p-group, then
Fitt(Aut G) - 0p(Aut G) 2 Inn G ’9 1.

2332;: If H is a group, Fitt(H) ==C> Z O (H). Con-
sequently Fitt(Aut G) 2 Op(Aut G), which contiiig‘the nontrivial
normal p-group Inn G. Let B = Stab(G 2 Z 2 l) and A the
direct complement of Op(Aut G) in Fitt(Aut G). Since A and
B comnute [B,A,G] = [1 ,c] = 1. By 3.9 [11,13,131 5 [2,13] = 1. The
Three Subgroup Lemma gives [G,B,A] = 1. That is, the p'—group A
centralizes [G,B] which contains 01(2). By Gorenstein [9, page
178], A must centralize Z. Because A is a p'-group contained
in the p-group Stab(G z z z 1), A = 1 and Fitt(Aut G) = Op(Aut G) .

Theorem 3.12. If G is an abelian p-group, not elementary
abelian of order 4 or 9, then Fitt(Aut G) = Op(Aut G) X B where
B is a cyclic subgroup of Z(Aut G) and \B\ = p - 1.

2332;; In [7], Hightower has shown that if G is an abelian
group of exponent pt not elementary abelian of order 4 or 9, then
\Fitt(Aut G)‘ = (p-l)pn for some integer n. Since G is abelian,
x a xi, (i,p) = 1, gives (p-l)(pt-l) automorphisms in the center
of Aut G. By restricting to <g> of order pt we may find that
power automorphism of order p-l.

In 1.7 we saw that [G,A] s Fitt G whenever A E It. Thus
it is not surprising that a condition about Fitt G might enable

us to say something about v G A1 and closed stability groups.

43

Theorem 3.13. If Fitt G is purely nonabelian then
Fitt(Aut G) is a closed stability group.

2522;; Let p # q, be two primes. By 3.8, Fitt(Aut G)
stabilizes G 2...2 2. Since Oq(G) is characteristic in G and
Z(0q(G)) 2 Oq(G) F\Z(G), Fitt(Aut G) must stabilize
Oq(G) 2...2 Z(Oq(G)). Thus Op(Aut G) induces a p-group of auto-
morphisms on Oq(G)/Z(Oq(G)) that stabilizes the series
Oq(G)/Z(Oq(G)) 2...2 Z(Oq(G))/Z(Oq(G)) = I; Since this stability
group must be a q-group Op(Aut G) centralizes Oq(G)/Z(Oq(G)).
0q(G) must be purely nonabelian because Fitt G is purely non-

abelian. By [2, Corollary 2], c (Oq(G)/Z(Oq(G))) is a

Aut Oq(G)
q-group. 0p(Aut G) is a p-group which induces a group of auto-
morphisms on 0q(G) which centralize 0q(G)/Z(Oq(G)). Thus
Op(Aut G) must centralize Oq(G). Since Op,(Z(G)) the p -
complement of Z(G) is contained in H O (G), O (Aut G)

q‘p q P
centralizes Op,(Z(G)). Op(Aut G) induces a p-group of auto-
morphisms on the p-group Z(G)/Op,(Z(G)), which by 2.1 must
stabilize a series Z(G)/Op,(Z(G)) 2...2 op,(Z(G))/op,(2(c)) .. '1'.
Op(Aut G) will then stabilize the preimage of that chain and con-
sequently the chain, G 2...2 Z 2...2 0p.(Z) 2 1. Let
A = 0p(Aut G) 4 Aut c. By 1.10.v, X = Stab(y G A1) 4 Aut G.
l.10.vi says that n(A) = n(A). A: is a normal p-group containing
A = 0P(Aut C) so Ah= Op(Aut G) is a closed stability group. In
1.20 we showed that the product of closed stability groups of

relatively prime order is a closed stability group, so

Fitt(Aut G) = H0p(Aut G) is a closed stability group.

44

Lemma 3.14. Let B = Stab(G 2 Op.(Z) z 1) and 2 be
the center of G. If Exp(0p,(Z)) divides Exp clop'(z)c'
then [G,B] = Op,(2) , and B is a closed stability group.
I_’_r_oo_f_: Since G/Op.(Z)G' is abelian.Exp G/Op.(Z)G' is
the product of the orders of the largest cyclic summand of each Sylow
of G/Op.(G)G'. Exp Op,(Z)\Exp G/Op,(Z)G' means that each cyclic direct
summand <y>' of a q—Sylow, q # p, of Op.(Z) has order less than
or equal to a cyclic direct summand <x>’ of the q-Sylow of
G/Op,(Z)G'. Thus there is a homomorphism f from <x> onto <y>

that may also be thought of as a homomorphism f: G/Op,(Z)G' into

0p,(Z).
As in 1.26 we let a: x a xxf for all x E G and see that
a 6 Stab(G 2 Op,(Z) 2 1) and [G,q] = <y>. Thus [G,B] contains
any cyclic direct summand of the q-Sylow of Op,(Z), and consequently
[G,B] 2 Op.(Z). [G,B] s Op,(Z) because B = Stab(G 2 0p,(Z) 2 1).
Thus [G,B] = Op,(Z) and B is closed.

Lemma 3.15 (B.I. Plotkin). Let F act on G, F = <A,B>,
B stabilize G 2 H 2 1, A normalize B, A stabilize a chain
G 2...2 H, H s Z(G), [B,A,...,A] = 1. Then A stabilizes a chain
[G,B] 2...2 l.

2529;; See Plotkin [12].

Theorem 3.16. Let 2 be the center of G. If Exp Op,(Z)

divides Exp G/Op,(Z)G' then Op(Aut G) is a closed stability

group.

 

Proof: Let A = Op(Aut G). By 3.8, A stabilizes a series
G 2 G1 2...2 Z. Z/Op,(Z) is a p-group acted upon by the p-group
A. A therefore stabilizes a chain Z 2...2 OP,(Z). Now we check

the conditions of 3.15 with B = Stab(G 2 Op,(Z) 2 l) and

45

H = Op.(Z). Note that BIQAAut G since B is the stability group
of a characteristic series. A stabilizes a series G 2...2 H = 0p,(Z),
H = 0p,(Z) 5 Z. [B,A,...,A] = 1 because .A s Fitt(Aut G). Finally,
3.15 says that A stabilizes a chain [G,B] 2...2 1. By 3.14
[G,B] = op,(2). Thus A stabilizes G 2...2 z 2...2 op,z 2...2 1.
By l.10.v, A-unt G. {p} = n(A) = n(A) by 1.7. Thus A is a
norml p-group of Aut G, i.e. A S 0p(Aut G). Op(Aut G) = A s A
by 1.10.1, so Op(Aut G) is a closed stability group.
Corollary 3.17. Let 2 be the center of G. If Exp 2
divides Exp G/Z'G' then Fitt(Aut G) is a closed stability group.
2329;; For an arbitrary prime p and Op.(Z), the p
complement of Z we have: Exp Op.(Z) divides Exp Z which
divides Exp G/Z-G' which divides Exp(G/Op,(Z)G'). By 3.16,
0p(Aut G) is a closed stability group. Since p was arbitrary
and the product of closed stability groups of relatively prime order
is again a closed stability group, Fitt(Aut G) = H Op(Aut G) is a

P
closed stability group.

CHAPTER IV

QUATERNIONS AS CLOSED
STABILITY GROUPS

In my search for examples of stability groups, I had trouble
finding a 2-group G and series 3 so that Stab(s) was isomorphic
to Q, the quaternion group of order eight. I even conjectured that
Q could not be a stability group. Since an arbitrary series s
is difficult to work.with, I first worked on the conjecture for a
normal series (each Gi <16). Toward the end of a long and rather
messy proof, the following two examples arose. The second is an
example where Q is a stability group but not a closed stability.
With the addition of the condition of closed stability group to my
conjecture I was able to prove 4.1.

Let G and s be defined by

8
G=<a,b‘a =b 331,3 =a>
and

2 2
s: G 2 <3 ,b> 2 <a -b> 2 1 .
We obtain 2 automorphisms Wsm E Stab(s), namely

W: a ~»a3 ¢z a -oa

b—obaa b—Obaa

Each one has order two. Stab(s) is not Q, because Q has only one

element of order 2.
46

47

2

If G - «,b‘a16 = b B 1, ab = a9> and

2 2
s: Gz<a ,b>2<a °b>2l then

(i) Q°'Stab(s)

. - 4 8

(ii) s:Gz<a,b>2<a>21

(iii) Q 2Q -

Theorem 4.1. If G is a 2-group, the quaternion group Q
is not a closed stability group for a normal series of G.

Proof: Assume Q = Stab(G = G 2.C 2...2 G = 1) where

0 1 n

Gi'VGQi and each G 46. Let A=Stab(G26

i 2 l) and

1
B = Stab(G 2 Gn_1 2 1).

The proof is carried out in a series of steps and cases.

(i) A FIB = Z(Q)-

Since [G,B] s and, [G, A [13, Q] s [Gn_1, Q]. Q fixes
Gn-l pointwise so [G, A GAB, Q] = 1. Since [Q, G] s G1 and G1
is fixed pointwise by A, we have [Q, G, A n B] = l. The Three
Subgroup Lemma gives us, [A FIB, Q, G] = 1. Since [A FIB, Q] acts
trivially on G, LA FIB, Q] = l and A FIB S.Z(Q). Z(Q) is cyclic
of order 2, so all that remains of (i) is to show that A 0‘3 2 1.
Let M be a maximal and therefore normal subgroup containing 61'
Since Gn-l is normal, we have Gn-l n Z(G) 2 1. Thus, there is

a nontrivial f E Hom(G/M, Gn-l FIZ(G)). By 1.26, a x a xxf is

f'
an automorphism. of fixes M pointwise and [G, of] is the image
of f which is contained in Gn—l n Z(G). Thus 1 # “f E A FIB.
(ii) G/Gn-lc' and G1 0 Z(G) are nontrivial cyclic groups.
By 1.12, A and B are abelian subgroups of Q which must,
therefore, be cyclic. Theorem 1.26 says that Hom(G/GIG', G1 fl Z(G))

and Hom(G/Gn_1G', Gn 1 fl Z(G)) are isomorphically contained in A

48

and B reapectively. Since A and B are both cyclic,

Hom(G/GIG', G1 n Z(G)) and Hom(G/Gn_1G', Gn_1 n Z(G)) are both

cyclic. By 1.27, G/Gn_1G' and G1 FIZ(G) are cyclic.

(iii) Either G/GIG' or Gn—l n Z(G) is cyclic of order
two.

In (1) we saw that

1 ‘Hom(G/M, Gn n Z(G)) s Hom(G/GlG , Gn_1 n Z(G))

-1

By 1.26, Hom(G/GIG', Gn—l fl Z(G)) is isomorphically contained in
A FIB. Since A FIB is cyclic of order two, 1.27 says that either

G/GIG' or G n Z(G) is cyclic of order two.

n-l

(iv) G is nonabelian.
Assume G is abelian. G is not cyclic since the auto-
morphism group of a cyclic group is abelian and thus could not con-

tain Q. By (iii), 6/61 or G is cyclic of order two. By (ii),

n-l

G and G/G
l n

G1 is cyclic of maximal possible order and we see that

G = <y> X <x>, where <x> = Cl. By 1.24,

_1 are cyclic. If G/G1 is cyclic of order 2, then

Q = Stab(G 2 <x>sz...2 1)

=* Stab(<x> x <y> 2 <x> 2 1) :IStab(G1 2 62 2...2 Gn = 1).
Since Q is nonabelian, n 2 3 by 1.12. By 2.27,
Stab(<x> x <y> 2 <x> 2 l) 2 1
and
Stab(G1 2 G2 2...2Gn = 1) 2 Stab(G1 2 G2 2 1) 2 1.

This is a contradiction since Q is not a semidirect product of

49

nontrivial subgroups. If Gn-l is cyclic of order 2 we see that

there is an x 6 G\Gn that has maximum possible order because

-1
G/Gn-l is cyclic. <x> is therefore a direct summand and
G = <x> Xi<y> where <y> = G . Again apply 1.24 to obtain

n-l

Q = Stab(G 2...2 1)

=- Stab(<x> x <y> 2 <y> 2 1)] Stab(G/On 2 G1/Gn 2...2 G /Gn_ =1)

-l -1 n-l 1

the desired contradiction. Thus we may assume that G is nonabelian.

(v) G/Gn is cyclic and either G/G1 or Gn-l fl Z(G)

-l

is cyclic of order 2.
By (ii) G/Gn_IG' is cyclic and since Q, the Frattini sub-

group, contains 6', G/Gn-IQ is cyclic. There exists x E G\Gn_1§

such that G = <x, Gn-l’ §>. Since Q is the set of nongenerators,

G = <x, Gn- >1 and G/Gn-l is cyclic. Since G/Gn-l is cyclic,

1

G' S'Gn-l 5 G1. The remaining conclusion follows from (iii).

(vi) CG(Gn_1) = [G, Q]Z(G) is an abelian group, and
[[G, Q]\ 2 8.
I o o o
Since G s Gn-l’ CG(Gn_1)/Z(G) is isomorphic to a group

of inner automorphisms in the cyclic group Stab(G 2 GD 2 l).

-1

(Gn-l) mod its center is cyclic so CG(Gn ) is abelian. Since

CG -1

GB 1 <6 and Q fixes Gn 1 pointwise, 1.13 says that

c .B ='
[G, Q] s G(Gn-l) ecause G/Gn 1 <X> for some x E G and

G is fixed pointwise, every a E Q is completely determined

n-l
by xa = x[x,a]. Thus there are at least eight different [x,a]'s.

[G, CG(Gn-l)’ Q] s 10', Q] = 1

and

50

1Q, 6. CG(Gn-l)] = [16. Q]. cG(Gn_1)] s [CG(Gn-1)’ C (Gm-1)] = 1.

G

so the Three Subgroup Lemma says that [CG(Gn_1), Q, G] 1. By

3.6 = [ , Q], so [_ , Q] = 1. This
"[CGmn-l) ’Q] 1'{ZGmn-l) rEGmn-l)

is equivalent to:

CG(Gn 1)/Z(G) is isomorphically contained in the center of Q .

[G,Q] G Z(G) or else we would have G/Z cyclic and G would be

abelian. Thus

IG,Q|Z(G) CG(Gn-l) c: Z(Q)

1 f Z(G) ‘ Z(G) 2

Since Z(Q) is cyclic of order 2, [G,Q]Z(G) = CG(Gn ) and Z(Q)

-l
is the set of inner automorphisms induced by CG(Gn_1).

Case I. Suppose that one of the cyclic factors Gi/Gi+1’
i = 0,1,2,...,n-2, has order greater than two. Thus

'IG/Gn = 2t > 4.

-1I

Since CQ(G/Gn = B = Stab(G 2 GD 2 l) is abelian, we

-1) -1

have that

1 $ Q/B c5. Stab(G/G“ 2 Gl/Gn 2...2 G /G )

-1 -1 n-1 n-1

Since one of the factors is larger than 2 and G/Gn is cyclic,

-1
2.17 says

1 fl Q/B C <5Ij> .
Any proper factor group of Q is elementary abelian, so

2t-3
1 s Q/B g 01(51) = <(51) >

51

and B must be cyclic of order four. Let B = <B>. If Q. indicates
the induced automorphisms, [G/Gn 1, Q] = [G,Q]/Gn 1 is cyclic of
3 - -

I:-
order two because (51) centralizes G/Gn mod its cyclic group

—1
of order two. This coupled with the fact that the series is closed
says that the series is G 2 LG, Q] 2 [G, Q, Q] 2 1 with
IG/LG, Q]I 2 4 and I[G, Q]/[G, Q, Q]I = 2. From part vi and an

isomorphism theorem we have

1cm .. Law 3 CG(Gn-1)
[G,Q] n Z(G) Z(G) Z(G)

 

is cyclic of order two. Since \[G,Q]\ 2 8, \[G,Q] n Z(G)] 2 4.

[c.Q; n Z(G)
[G.Q.Q] FIZ(G)

which is cyclic of order two. \G/Gl‘ 2 4 so v implies

is isomorphically contained in [G,Q]/Gn 1

'Gn-l FIZ(G)| = 2. This in turn forces ILG,Q] n Z(G)] = 4. By
(ii) G1 FIZ(G) is cyclic with some generator 2. Since

.A a Stab(G 2 G 2 l) is an abelian subgroup of Q, \A| s 4. Let

1
x be an element of G that maps onto a generator of G/Gn-l' By
1.26, a: x --9 X2

1, is a member of 'Stab(G 2G1 2 l)

and since its order is four, <a>’=.A. Since IA fl‘BI = 2, we see

g a g for all g E G

that Q = <a,5>. Since (3),: Stab(G 2 [G,Q,Q] 2 1) and
[6.0] £ 2(6).
i i
xB = x(xB) for some xB E [G,Q,Q]\Z(G)

Since ‘5‘ = 4,

[G, <g>] = <xa> is a normal subgroup of order 4.

52

<z,xa> is a normal subgroup. Since a and B centralize

Gl<z,xe>3 <2,x8> 2 [G,Q]. The other inclusion is trivial so

t-l
2
<z,xB> = [G,Q]. We now have C = <x, [G,Q]> = <x,z,x8>. x is
a member of the abelian group [G,Q] because \G/[G,Q]\ = 2t-1.
Zt'l t 2 2
x E [G,Q,Q] Since IG/[G,Q,Q]I = 2 . Since a = B , we see
that
2 2
z = x .
5

Since [G,Q] n 2(a) = <z> and 22 = x: e [G,Q,Q],

Because 2 was chosen as an arbitrary generator of G1 fl Z(G), we

may assume

The inner automorphism fix induced by xa must have order two

8
because xB é Z(G) but x: 6 Z(G). Since [G,Q,Q] was assumed

to be normal and [G,Q,Q] is abelian,

n E <g> = Stab(G 2 G 2 1)
x n-l
B
2
Because n has order two, n = B .
x x
S B
3 x3
x x x = x
B
2
.xe
2
= xz
2

53

Cancelling x 's x and

B

we see, x xxB =

8)2 = x2 .

(xx

The following calculation shows that

tion to the fact that Q is nonabelian.

x06 (xz)8

. c-1
= (xx2 )

B

t-l
= xx (XX )2

B B 2
2 2“
— xxa((xxe: :
= xxe(X2)2

2t-l

= X
xxB

= xsz

xBa

= (xx )0

B

= XZX

= xx 2 .
8

d8 = aa, a contradic-

is nonabelian and by vi

and

Case II. Suppose that each of the factors Gi/Gi+l’
i = 0,1,2,...,n-2, has order two. G
Cc(Gn_1) = [G,Q]Z(G) is abelian. \G/[G,Q]I = 2

G 2 [G,Q]Z(G) 2 [G,Q] so

16.0] = [6.012(6) = cG(cn_1>

and

[G,Q] 2 2(G)

54

‘ : Z G = 2, s w
By v1, [CG(Gn-l) ( )] o no

[[G,Q] : Z(G)] = 2 .

[G,Q,Q] 2:1 or else the series would have length two and Q would
have to be abelian. [G,Q,Q] n Z(G) # [G,Q,Q] or else we would have

G/Z(G) cyclic and G abelian. Thus

16.0.01 ... LG. ...0._01_i_iz G C L316

1 ‘ [6.0.0] n Z(G) Z(G) .. Z(G)

 

Since [G,Q]/Z(G) has order two, Iég’g’gjl O Z(G) has order two and

[6.0.0]Z(G) = [6.0]. By (ii), Z(G) = [G,Q] n Z(G) is cyclic and
[G : Z(G)] = [G : [6,0]][[G.0] = Z(G)] = 4 .

Case II-a. Suppose G has no cyclic subgroup of index two.
We know that there is a cyclic normal subgroup of index four, namely
<a> = Z(G). Burnside, [4, page 138], says

-2 m-3
2m 2 2
G = <a,b,c‘a e Z(G), a 2 C

II
...a
II
0‘
ll
0
0"
II
U
93
V

One may calculate that u and v, defined by

p: a a a and v: a a a
b a c b a b
2m-4
C a b c a a bc ,

are automorphisms and

V
<a,b> 9. <a,c> —. <a,bc> .

From the Appendix we see that m 2 4 because the quaternions

are not a stability group for either the quaternion or dihedral group

55

of order eight. Since G/Z(G) is elementary abelian and

[[G,Q] : Z(G)] = 2, the only possibilities for [G,Q] are

<a,b>3 <a,c>3 and <a,bc>u We may assume that [G,Q] = <a,b> or
else [G,QY] = [G,Q]Y = <a,b> for some conjugate Q.Y of Q in

Aut G.

 

C
2(6) . ____ I746) 6 [6.0.01‘

‘2<6) n16.0,0]l I-G—I
Z(G)

 

Z(G) is cyclic and therefore Z(G) FI[G,Q,Q] = <a2> is the only
subgroup of index two. [G,Q]/<az>‘* (<a>'X <b>)/<az> is elementary
abelian of order four. Since <a,b> 2 [G,Q,Q] 2 <a2>’ and has
[[G,Q,Q] : <az>] = 2, [G,Q,Q] must be one of the three subgroups
<a>3 <az,b> or <ab>. [G,Q,Q] = <a> = Z(G) contradicts G non-
abelian and G/[G,Q,Q] cyclic. Neither G/<ab> nor G/<az,b> is
cyclic contradicting G/[G,Q,Q] is cyclic.

Case IItb. Suppose G has a cyclic subgroup of index two.
Z(G) has index four or else G is abelian and the Appendix again
shows that \GI 2 16. Thus IZ(G)\ 2 4. By Huppert [8, page 91],

the only groups of order ZM-1 with a cyclic subgroup of index 2 are:

Dihedral, <a,bIab = a-1, a2“ = b2 = l>;

Generalized Quaternion, <a,b\ab = a-l, a2n = l, b2 = a2n-1>,
Quasidihedral, <a,b‘ab = _1+2I1-1, a2“ = b - l>,

and <a,b‘ab = a1+2n-1, 2n = b2 = 1)

56

One may calculate that in all but the last case, \Z(G)\ = 2.
Now \G| = 2n+1 2 16, so n 2 3 and n-1 2 2. We see that
<82> s Z(G) and Z(G) cannot be larger or G would be abelian.
2
Thus Z(G) = ¢ >.

By Case 11, we have the following diagram,

11 -—0

[G.Q] [G,Q.Q]Z(G)
2 2

«12> = 2(6) [6.0.0]

2
[G.Q.Q] 0 Z(G)

where the numbers indicate the indices. We notice that
2
[Z(G) : [G,Q,Q] n Z(G)] = 2. Since Z(G) *3 (a > has only one sub-

group of index two

[6.0.0] n 2(6) = «‘3 .

Since G/Z(G) = GI¢2> is elementary abelian and
[G,Q] 2 <a2>, [G,Q] is one of <a2,b>, <a>, or cab). [G,Q] is
not cyclic or else [G,Q,Q] and Z(G) would both be of index two
and therefore equal. Thus [G,Q] = <a,b>. Using the same elementary
abelian argument again, we see that since <a2,b> 2 [G,Q,Q] 2 <a4>,
[G,Q,Q] must be one of <32), <b,a4>, <32b>. We rule out <32>
since [G,Q,Q] i Z(G). Examination of Stab(G 2 <az,b> 2 <b,a4> 2 1)
gives two automorphisms of order two:

+2“'1
Tibia-+81 and ¢:a-oab

b-ob b-Ob.

i‘lllll‘llllll‘l‘. '1} II

 

57

4
Thus Stab(G 2 a2,b> 2 <b,a > 2 1) £ Q as would be the case if

<b,a4> = [G,Q,Q]. The only possibility left is that
2
[G.Q.Q] = <61 b> -

If [G,Q,Q,Q] i 1, then <a4>’ and [G,Q,Q,Q] are two sub-
groups of index two in the cyclic group <azb>. This gives
[G,Q,Q,Q] = <34>5 Since G/[G,Q,Q,Q] is cyclic and <34) S Z(G),
G would have to be abelian. Since G is nonabelian
[c.QsQJl] = 1-

Thus Stab(G 2 <az,b> 2 <azb> 2 1) = Q and Q is closed.
The two examples in the beginning of the chapter show that

n i 3 or 4. If B E Q then

2. c -
y: a —iaa 1bJ ... n 1

+ 2
b-b(a2b))‘=b1 "a A j =o,1.

Calculation of the effect of W on the relations of G shows that

W must have the form

t: a —.aa2i'b‘1 x1,j = 0,1
n-l A _
b—ob(a2 )1 i=0,1,2,...,2“1-1.

2 2
(a b)w = a b limits us to eight possibilities.

2n-l 2n-2
1 , a 4 aa a -oaa
G , 2n-l
b ~‘b b a ba
-2 n-2 n-2
-2“ -2
a -o aa a ... aa2 b a -+ as b
211-].
b ~sba b a b b a b
2n-l
a ~»aa b a -.ab
2n-l 20-1

b ‘°b8 b —.ba

58

We see that each of these eight possibilities centralizes
-2
2n
G/<a ,b>- which has order

]<a>] = 2n = 2n-2
2n-2 E2
I<a >1
2n-2
Since n 2 5 we see that [G,Q] is contained in <a ,b>- a

subgroup of index 2“”2 > 2.
Thus Q is not a closed stability group for a normal series.
Corollary 4.2. Let G be a 2-group and Q the quaternion

group of order 8. Then
(i) Q1? 82(Aut G), the 2-Sylow of Aut G.

(ii) Q #- c (6/0).

Aut G
(iii) Q 4- 02(Aut G) the 2-Sylow of Fitt(Aut G).

(iv) Q is not a K-type stability group.

Proof: By 2.1, 1.15, 2.8, and 2.5 respectively, the

groups S2(Aut G), C (6/0), 02(Aut G) and any K-type stability

Aut G
group are closed stability groups. Since Inn G normalizes each
one, l.10.vii gives that each of the four automorphism groups is a
closed stability group for a normal series and therefore not
isomorphic to Q.

Corollary 4.3. There is no group G with Aut G °'Q, the
quaternion group of order eight.

2522;: .Assume to the contrary that Aut C ”'Q for some
group G. G/Z “'Inn G s Aut G, so G is nilpotent. Let Sp be
a p-Sylow for an odd prime p. If p2 divides ]G], we see in
[1] that p divides ]Aut G]. Thus ]Sp] = p or 1. If

]Sp] = p, Aut(Sp)‘* a(p-l), a cyclic group of order p-l, and is

59

a direct summand of Aut G 9' Q. Since Q has no cyclic direct
summands, Sp 8 1. We are now left with the 2-group G and

Q °' Aut G = 82(Aut G) a contradiction to 4.2.i.

APPENDIX

APPENDIX

 

 

 

In the following examples A will indicate that A is a

closed stability group. G means A is a stability group but not

closed. A series of arrows such as

indicates that C is the chSure of B and A.

 

Example 1

The quaternion group of order 8 is given by

2 2 4 -
Q = <x,y]x = y , x = 1, xy = x 1>, and has the following subgroup

lattice. Q

\

<y> <x> <xy>

/

<X

 

Aut Q is the set of mappings taking x to one of the six
elements of order four (x, x3, y y3, xy, x3y) and y to one of

the four remaining elements not in the subgroup generated by the

image of x. Aut Q “'Sym (4).
60

61

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P1
A1 A2 Inn Q = 02(Aut Q) A3
1
Stability Groups
2
P1 =Stab(Q 2<x>2<x >2 1)

P = Stab(Q 2 <y> 2 <x2> 2 1)

2
P = Stab(Q 2 <xy> 2 <x > 2 l)
Inn Q = Stab(Q 2 <x2> 2 l)
A = Stab(Q 2 <x> 2 1)

A2 = Stab(Q 2 <y> 2 l)
A = Stab(Q 2 <xy> 2 1)
Pi is dihedral of order eight and A1 is cyclic of order

four for i = 1,2,3. Each stability group is closed with s being

the series listed.

62

Example 2
The dihedral group of order eight is given by

2
D = <x,y]x4 = y = 1, yxy = x3>' and has the following subgroup

 

4
lattice.
<yix 2) 22xy>
3
(y) <x2 y> <x y>

\]//

By finding all relation preserving onto mappings we see that

4»
Aut D4 = 41,5](1 = 92 = l, as = a >=-D where

 

 

 

 

 

 

 

4
a: x a x 83 x a x
Y * YX Y 4 Y
D
Aut 4
2
(8.01 > <a «12,01 38>

 

 

 

 

 

 

.f.\3s

 

 

 

 

 

 

63

Stability Groups

2
Aut D = Stab(Da 2 <x> 2 <x > 2 1)

2 2
(Bad > Inn G = Stab(D4 2 <x > 2 l)

2 2

= Stab(D4 2 <3: ,y> 2 or > 2 1)
2 2

= Stab(Dl‘ 2 <x .xy> 2 <2: > 2 1)
2

<g> = Stab(D4 2 <x ,y> 2 1)

2

= Stab(D4 2 <x ,y> 2 <y> 2 1)

2 2
= Stab(D4 2 or ,y> 2 <3: y> 2 1)

2
Stab(D4 2 <xy,x > 2 l)

2
<a8>
= Stab(DA 2 <xy,x2> 2 <xy> 2 l)

2
= Stab(D4 2 <xy,x >,2 <x3y> 2 1)

<g> = Stab(D4 2 <x> 2 1)

Closed Stability Groups s
Aut D4 D42a>2<x2>21
2 2
<0! ,e> D4 2 <x > 2 1
<g> D4 2 co 2 1
Example 3

Let G = <x> X <y> where ]x] = 4 and ]y] = 2.

/|G\,

<xy> <x> <x .Y>
\]2/ 2 \
or > <x y> <y>

W

1

(,4

An automorphism of G takes x to any one of the four

2
elements (x, x3, xy, x3y) of order four and y to y or x y.

Aut G is isomorphic to

D4 and is generated by

 

 

 

 

 

n: x a xy 3: x a x
2 2
Y r x Y Y * X Y
Aut G
2 4\ 2
<n ,B> <n> “II .IIB>

 

 

 

<B> @ flag? 46>

 

 

 

 

 

 

 

 

 

 

 

-l/

l

 

 

 

 

Stability Groups

2 2
AutG=Stab(G2<x .y>2<x>21)

Stab(G

2
<3,“ >
= Stab(G

= Stab(G
<fila> = Stab(G
<B> = Stab(G

<02.HB> = Stab(G
= Stab(G

= Stab(G
<ne> = Stab(G

(“38> = Stab(G

2
2<x>2<x>21)
2
2<x>21)

2
2 <Xy> 2 <x > 2 l)

N

«>2 1)

N

<xy> 2 l)

2
2o: ,y>21)
2 2
2<x ,y>2<xy>2l)

2
2<x ,y>2<y>21)
2 <y> 2 l)

2 <x2y> 2 1)

65

Closed Stability Groups 3
2 2
AutG G2<x,y>2<x>21
2 2
<1I,5> G2<x>21
3 2
<‘IIB> Gz<xy>21
(na> G2<y>2l
2 i 2
<n,ne> G2<x.y>21
Example 4

The nonabelian group G of order 33 and exponent 32, given by

<a,b]a9 = b3 = 1, ab .= a4> ,

has the following subgroup lattice.

G
M3
<ab> <a> <ab > <a ,b>
W 2
<a > <a b> <b> <a b >
1

Aut G has a normal 3-Sylow generated by

3 3 3
P =<a.a.v]0 =6 =v =1. aB=Bavsav=Yas BY=Y5> which

is the nonabelian group of order 33 and exponent 3.

a: a dish a: a a a y: a a a

b—oa3b b—ta3b b-sb

66

 

 

 

 

 

 

Aut G
P
/\\\
2
B>”<Y> Inn G = <B><y>

 

 

 

 

 

 

<e> @ ¢

/

 

 

 

 

Stability Groups

P = Stab(G 2 a3,b> 2 <a3> 2 1)

2
(By) = Stab(G 2 <ab > 2 1)

<a3,b> 2 1)

Stab(G

N

2
<0 B><y>
= Stab(G 2 <33,b> 2 <b> 2 1)
= Stab(G 2 <a3,b> 2 413b> 2 1)

2
= Stab(G <33,b> 2 <a3b > 2 1)

N

Inn G = Stab(G <33> 2 1)

N

= Stab(G

N

<a>2 <a3> 2 1)

2
= Stab(G <ab > 2 <a3> 2 l)

N

= Stab(G 2 <ab> 2 <33> 2 l)

<6> 8 tab (G

N

<a>2 1)

2
<8 y) = Stab(G

N

<ab> 2 l)

67

Closed Stability Groups 8
3 3
P Gz¢,b>2<a>21
2 3
<ae><y> G2<a,b>21
Inn G G 2i<33> 2 1
Example 5

The nonabelian group of order 27 and exponent 3 is given by

G =<.a,b,c]a3 =b =c = 1, ac =ca, bc=cb, ab =bac>

and has the following subgroup lattice.

 

 

6
N1 N2 N3 ,N4
H1 H2 H3 H4 H5 H6 (G) H7 H8 9 H101111 H12
1

Each N1 is elementary abelian of order nine and Hj is

cyclic of order three, i 8 1,2,3,4, j = 0,1,2,...,12.

68

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Aut G
AX
P1 P2 P3 P4
K1 K2 Inn G K3 K4
1

Each 3-Sylow P1 is isomorphic to G and Ki is

elementary abelian of order nine, for i = l,2,3,4.

Stability Groups

P1=Stab(G2N12221) i=1,2,3,4
Inn G = Stab(G 2 Z 2 1)
K1 = Stab(G 2N1 2 1) i = l,2,3,4
= Stab(G 211i 2 H12 1) j = 31-2, 31-1, 3i
Closed Stability Groups S
Pi G 2N1 2 Z 2 l i = 1,2,3,4
Inn G G 2 Z 2 1

K1 6211121 1 1,2,3,4

B IBLIOGRAPHY

10.

11.

12.

13.

BIBLIOGRAPHY

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of a Finite Group. American Mathematical Monthly, 59(1952),
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Adney, J. and Yen, T. Automorphisms of a p-Group. Illinois
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Blackburn, N. Automorphisms of Finite p-Groups. Journal of
Algebra, 3(1966), 28-29.

Burnside, W. Theory of Groups of Finite Order (New York: Dover
Publications, Inc., 1955).

Hall, P. Some Sufficient Conditions for a Group to be Nilpotent.
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Hall, P. and Hartley, B. The Stability Group of a Series of
Subgroups. Proceedings of London Mathematical Society,
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Hightower, W. Some Subgroups of the Automorphism Group of a
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Huppert, B. Endliche Gruppen I (Berlin: Springer Verlag, 1967).

Gorenstein, D. Finite Groups (New York: Harper and Row, 1968).

Kaloujnine, L. Ubergggwisse Beggichunggn zwischen einer Gruppe
und ihren Automorphismen. Berliner Mathematische Tagung,
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. 0b Odnem Atmoshenii Galun V teorie grupp. Ukrainian
Matematicheskii Zhurnal, 11(1959), 38-50.

Plotkin, B.I. Some Properties of Automorphisms of Nilpotent
Groups. Doklady Akademiia Nauk SSSR, 137(1961), 1303-1306.

Schmid, P. Uber die StabilitEtsgruppen der Untergruppenreihen

 

einer endlichen Gruppe. Mathematische Zeitschrift,
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Schmid, P. Uber die Automorphismengruppen endlicher Gruppen.
Archive der Mathematik, 23(1972), 236-242.

Scott, W. Group Theory (Englewood Cliffs, New Jersey: Prentice
Hall, 1964).

Shoda, K. Automorphismen Abelscher Gruppen. Mathematische
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Suprunenko, D.A. Soluble and Nilpotent Linear Groups,
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