H :AUTOMORPHISMS

Thesis for the Degree of Ph. D.
MSCHEGAN STATE UNIVERSITY
JAMES RUSSELL WEAVER
1969

  

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Univemty

     

This is to certify that the

thesis entitled

H—Automorphisms

presented by

James Russell Weaver

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mathematics
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ABSTRACT

H-AUTOMORPHISMS
By

James Russell Weaver

The automorphism group of a group is one of the main topics
in the study of finite groups. If one is able to determine an auto-
morphism or an automorphism group of a group then it is conceivable
that some information about the group may be obtained. In this thesis
the existence of an H-automorphism for H a subgroup of group G
will be used to determine if the group factors into the product of
two proper subgroups. We will also consider the structure of a
group containing certain subgroups H which are T(G;H)-closed in G.

H. Liebeck calls an automorphism a an H-automorphism for H
a subgroup of G if g-lga 6 H for each 3 in G. Associated with
the set T of H-automorphisms is the multiplier subgroup of T,
which is the normal subgroup of G generated by the elements of the
form g-lga where a G T and g E G.

In chapter I we follow Wielandt and define a Subgroup H to
be A-intravariant in G if for each a in A there exists a g
in G such that Hag = H. Then we define a subgroup to be A-closed
in G if it is weakly closed in its normalizer in G and is A-
intravariant in G.

In general, the larger a subgroup H is, the easier it is to

determine if there exists an H-automorphism. We will let F(G;K) be

 

 

James Russell Weaver

the set of automorphisms of G which fix K as a set. Using these
concepts we show that if H is F(G;NG(H))-closed in G then the
set of NG(H)-automorphisms fix each conjugate of H in G.

In chapter II we will look at groups in which each subgroup of
a Sylow p-subgroup is weakly closed in its normalizer in S for each
prime dividing the order of G. We will see that such groups are
Sylow tower groups for the natural ordering of the primes. Using the
result just mentioned we shall see that a group G is cyclic if and
only if each subgroup of a Sylow p-Subgroup Hp is F(G;Hp)-closed
in G and Hp i G'(p) for each prime p dividing the order of G.

It is in the latter part of chapter II that we make the
assumption that H is A(G)-closed in the solvable group G with
[c: 14601)] a prime. If 1"(G;H)/1"(G;H) 0 1(a) is solvable with the
set of NG(H)-automorphism equal to the identity subgroup of the
A(G) then there exists a subgroup T of the A(G) such that the
following hold:
1) CA(G)(T) = T,
ii) T is the minimal normal subgroup of G,
iii) A(G) = I"(G;H)T where I‘(G;H) n T E <1>,
iv) |T| = [0: NG(H)]
v) If Go/T is a minimal normal subgroup of A(G)/T and

A0 = 1"(G;H) n so then 1"(G;H) = NA(G)(A0), |co| = |T||Ao|,

‘T' = p8 and lel = qB and p # q for p and q *primes.

In chapter III we use the existence of an H-automorphism and
a NG(H)-automorphism to derive the existence of complements and
supplements of a subgroup of G. We first show that a NG(H)'aUt°'

morphism whose order is relatively prime to the order of NG(H)G

 

 

 

 

James Russell Weaver

and acts as the identity on NG(H)G/H n NG(H)G where H is
F(G;NG(H))-closed in G is an H-automorphism.

Next we establish that if H is a self-normalizing subgroup
of G which has an H—automorphism which fixes no proper subgroup of
H containing HG then there exists a normal subgroup G* of G
such that G = G*H and G* n H = HG. This will be useful in deter-
mining if a group G is solvable.

Burnside has shown that if a Sylow p-subgroup S of a group
lies in the center of its normalizer then the group has a normal S
complement. We will call any subgroup H of G with this property
a B-subgroup of G. Suppose that H is an Abelian B-subgroup of G
which is a Hall subgroup of its normalizer. If M is a complement
of H in NG(H), a is a NG(H)-automorphism, the set of fixed points
of a intersected with M is l, and the order of the multiplier
Subgroup of <a> and |H| are relatively prime then G has a
normal H-complement and H s Z(NG(H))-

We finish chapter III with the result which was one of the
original reasons for looking at F(G;NG(H))-closed subgroups of a
group G. Suppose that H is a Hall subgroup of its normalizer F
in G. If M is a complement of H in F, M S PG, and a is a
F-automorphism fixing no proper subgroup of H containing HG then
there exists a unique normal subgroup G* 0f G SUCh that G* A G,
G = G*H and HG = G* n H.

In chapter IV we consider the set of elements in a group G
which induce P-automorphisms of a group G for P a Sylow p-subgroup

of G. We call this set 3. f is shown to be a group which has a

p-complement M. We then show that G = PGNG(M)-

 

 

 

 

 

H-AUTOMORPHISMS

BY

James Russell Weaver

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Mathematics

1969

 

 

 

 

 

 

T0 JUDY

ii

 

 

ACKNOWLEDGMENTS

I wish to thank Dr. J.E. Adney for the many hours which he
has spent with me during the preparation of this thesis. He was
always able to find time in his busy schedule to listen to an idea
and then to give some constructive criticism. I would also like
to thank Dr. W.E. Deskins for the guidance which he gave me in
Dr. Adney's absence and for the helpful suggestions he has given

me in the past two years.

 

 

TABLE OF CONTENTS

INTRODUCTION ................................................
CHAPTER I
Basic Definitions and Properties ............. . .......
1.1 H-automorphisms ..............................
1.2 A-intravariant Subgroups .....................
1.3 A-closed Subgroups ............... . ...........
1.4 F(G;NG(H))-closed and Equivalences ...........
1.5 An Example ...................................
CHAPTER II
Groups Whose Subgroups are A-closed or A-intravariant

2.1 Subgroups which are Weakly Closed in their
Normalizers or F(G;H)—closed in G ...........

2.2 A(G)-closed and A(G)-intravariant Subgroups ..

2.3 An Upper Bound for the Order of A(G) ........
CHAPTER III
H-Automorphisms and NG(H)-Automorphisms ..............

3.1 NG(H)-automorphism which are H-automorphisms .
3.2 H-automorphisms ..............................
3.3 NG(H)-automorphisms ..........................
CHAPTER IV
Inner H-Automorphisms ............................. ...
4.1 Basic Properties of H .......................
4.2 H and Supplements of HG in G .............
BIBLIOGRAPHY ................................................

APPENDIX ....................................................

15

17

19
26

30

33
33
35

37

43
43
45
49

50

 

 

 

 

 

LIST OF FIGURES

Figure l .......................................... 13

Figure 2 .......................................... 14

 

 

 

II.

Relations:

S

D E>

Ill

Operations:

G9

HX
f|H
G/H

[H.K]
[XJ]
[G : H]

<x|P>

{KlP}
lGl

 

 

INDEX OF NOTATION

Is or be a subgroup of
Is or be a proper subgroup of
Is or be a normal subgroup of

Is or be a proper normal subgroup of

 

Is isomorphic to
Is an element of

Is equal to

The image of G under the mapping 6
x-IH x

Mapping of H induced by f

Factor Group

Subgroup generated by all [h,k],

h E H, k E K.

The commutator of x and y

Direct product of groups

Index of H in G

Subgroup generated for all x such
that P is true

Set whose members are

The set of all x such that P is true

Number of elements in G

 

 

 

III.

Groups:

A(G)
z(G)
cG(H)
NG(H)
1(0)

6' (p)

<L>

Fa(G)

F(G;H)

A(G;H)

K(G;T)

 

 

Order of the element g
The inner automorphism of G induced

by g in G.

Automorphism group of G
Center of G

Centralizer of H in G
Normalizer of H in G
Inner automorphisms of G

Characteristic minimum subgroup K of

 

G such that G/K is an Abelian p-group
Identity subgroup of a group

Set of elements of G which are fixed
by the mapping a

The subgroup of the automorphism group
of G which leaves each element of H
fixed.

The symmetric group of degree n

The alternating group of degree n

The derived group of G

Core of the subgroup H in G

The set of automorphisms in A(G)

which leave H fixed as a set.

The set of automorphisms in A(G) such
that g‘lga E H for each g in G.
The subgroup of G generated by
elements of the form g‘1§“ for g

in G and a in T for T a sub-

group of A(G).

 

 

INTRODUCTION

In chapter I an H-automorphism is defined for H a subgroup
of G. The motivation for the study of the set of H-automorphisms
came from H. Liebeck [8]. He shows that if R is a subgroup of H
which is admissible with respect to the set of H-automorphisms

A(G;R), then the set of R-automorphisms A(G;R), is a normal subgroup

 

of A(G;H). Furthermore, A(G;H)/A(G;R) is isomorphic to the sub-
group of automorphisms in A(G/K;H/K) that can be extended to G
where K is the subgroup of G generated by elements of the form
g-lga for g in G and a in A(G;H). Many questions can be
and have been asked as to what the existence of an H-automorphism
tells us about the group G for particular choices of H. Many
interesting questions will go unanswered due to the scope of the
topic.

We first consider how the set of NG(H)-automorphisms act on
the conjugates of H in G. We see that the set of NG(H)-auto-
morphisms fix each conjugate of H in G if H is weakly closed
in its normalizer and each element of A(G) which fixes the nor-
malizer of H in G moves H to a conjugate of H. The set
of elements of A(G) which fix a subgroup R of G is denoted
by F(G;R). In general we will say that H is A-closed in G
for A a subgroup of A(G) if H is weakly closed in its nor-
malizer and each element in A moves H to a conjugate of H

in G.

 

 

 

In chapter II we see that a group G is cyclic if and only
if each Subgroup K of a Sylow p-subgroup Hp is F(G;Hp)-closed
in G and Hp é G'(p) for each prime dividing the order of G.
We also discuss what happens when the hypothesis are not quite as
strong.

In chapter III we switch our attention to the existence of
a H-automorphism or a NG(H)-automorphism for H a subgroup of G.
We find that if some knowledge of its action on certain subgroups
is known then the existence of a supplement of H in G can be
deduced.

Finally, in chapter IV we consider those elements of the
group G itself which induce H-automorphisms of the group G.

We will use this set to establish the existence of a supplement
of the core of H in G for a particular choice of H.

The reader is advised to consult the index of notation for
the identification of symbolic notations of groups and relations.
The appendix consists of the known theorems used in thiSchesis

which are listedmasrmhaooem A, Theorem B, etc..

 

 

 

gn

Chc

 

 

CHAPTER I

BASIC DEFINITIONS AND PROPERTIES

In this chapter we will look at the main definitions and dis-
cuss some of the basic properties associated with these definitions.

In this thesis all groups will be assumed to be finite.

1.1 H-automorphisms

 

Hans Liebeck [8] has proven a number of results about H-auto-
morphisms. He also introduced the notation which we will adopt.
Definition 1.1.1: Let H be a subgroup of the group G.
An automorphism a of G is called an H-automorphism if g_1ga E H

for each g in G. Denote the set of all H-automorphisms by A(G;H).

Proposition 1.1.2: If H s G then A(G;H) s A(G).

Ppppfz If a and y are in A(G;H) then
g-lguy = g_]'(gx)'Y = g_1gxy = xy for x and y in H. Therefore
A(G;H) s A(G).

Conversely; any subgroup T of A(G) can be regarded as a
group of H-automorphisms for various choices of H. We can always
choose H = G for A(G;G) = A(G).

Let G be a group with T a subgroup of A(G). Denote by
K(G;T) the <g-1gala E T, g E G>. We will call K(G;T) the multiplier
subgroup of T. Note that T may be a proper subgroup of the group
of all K(G;T)-automorphisms. For example, if

2 b

G = S3 =‘<a,b|a3 = b = 1, a = a2> and T is the subgroup of the

 

 

 

 

 

A(G) consisting of the identity and n

b then K(G;A(G)) = K(G;T) = <a>

and T < A(G). One can easily show, as many already have,

Proposition 1.1.3: If G is a group and T s A(G) then

K(G;T) A G.

Proof: Let g- ga be a generator of K(G;T). If y E G then
-1 -1 -1 -1 -1 a -1 -1 a -1 -1
(s 80')y = y 8 say = y 3 gay (ya) y = (8y) r'(sy) (y ya) which

is in K(G;T). Therefore K(G;T) A G.

Denote by F(G;H) the set of a in A(G) such that H? = H
for H a subgroup of G. P(G;H) is clearly a subgroup of the A(G)
which contains A(G;H). We will now show

Proposition 1.1.4: If H S G then A(G;H) A F(G;H).
Proof: Let a E A(G;H) and let y E F(G;H). If g 6 G

then
‘1 flow -1 Y- Y -1 v v
8 g = g (g h) = g gh = h for some h in H Hence

h.Y E H and y-hyy E A(G;H). Therefore A(G;H) A F(G;H).

Since A(G;H) A 1"(G;H); an obvious question to ask is, "What

relationship exists between F(G;H), A(G;H), and K(G;A(G;H))?" The

following will give some information in this direction.
Lemma 1.1.5: If H s G then K(G;A(G;H)) is admissible with
respect to P(G;H).

Proof: K(G;A(G;H)) is generated by elements of the form

g-lga where a E A(G;H) and g E G. If y E T(G;H), then
613“)" = (€04an = (gY)_1(sY)Y-1°’Y- Since A(G;H) A I‘(G;H).

(g 1ga)Y E K(G;A(G;H)) and K(G;A(G;H)) is admissible with reapect
to F(G;H).

Theorem 1.1.6: If H s G then there exists a normal subgroup
M of G such that K(G;A(G;H)) S M s H, F(G;H)/A(G;H) is isomorphic
to a subgroup of A(G/M), and M is admissible with reSpect to

 

 

 

 

T(C;H). M is also maximal with respect to being normal in G, being

a subgroup of H, and being admissible with respect to F(G;H).
Ppgpf: By 1.1.3 and 1.1.5 K(G;A(G;H)) is normal in G, is

a subgroup of H, and is admissible with respect to F(G;H). Let M

be a subgroup of G maximal with respect to these three properties.
Now define ¢ mapping F(G;H) into A(G/M) by

(3")(Y)¢ = (gY)c where c is the natural homomorphism of G onto

G/M and y E F(G;H). If g6 and ha are two elements of G/M

then (g°h°>(Y’¢ = (<eh)°)“”¢ = ((ghfl)” = ((gY)<hY)>° = (2")“(hY)” =

(ga)(Y)¢(ha)(Y)¢. It then follows that (y)¢ is an automorphism of

G/M for each y in F(G;H).

If v and B are in 1"(G;H) (g°)(YB)¢ = (3%)“ = ((gV)°)(9)¢ =
(ga)(Y)¢(B)¢. From this it follows that ¢ is a homomorphism of
P(G;H) into A(G/M).

The kernel of ¢ is {v 6 1"(G;Ii1)|(ga)(wa5 = go for each g
in c} = {v e 1‘(G;H)|(g'1)°’(g")° = 1" for each g in G'} =
{y E I‘(G;H)‘g-1gV 6 M for each g in G} = A(G;H). Therefore
F(G;H)/A(G;H) is isomorphic to a subgroup of A(G/M).

If H s G then we will need to know what happens to F(G;H)
Vand A(G;H) if we conjugate either by an element of A(G).

Proposition 1.1.7: If H s G and a 6 A(G) then F(G;H)a =
1‘(G;h°’) and A(G;H)“ =A(G;H°’).

1321;: Let y 6 Mom). If h°’ e a“ then (h"’)"01 = (hY)°'
which is an element of Hg. Hence F(G;H)a s F(G;HQ). 1

Conversely, if v e Home”) and h e H then hdYa—l = x‘m- = x

for some x in H. Therefore I"(G;H)01 = F(G;HQ).

 

 

 

 

 

The proof of the second part is as straightforward as the first.

-1
For x in G and y in A(G;H) we have that x lxa Ya =
-l
- -l
x 1(xa h)a = x xha = ha for some h in H. Hence A(G;H)a S A(G;Hy).
-1 _
If y E A(G;HQ) we have that x-lxaya = x-1(xaha)a = h for some

h in H. Therefore A(G;Ha) =A(G;H)°’.
1.2 H-intravariant Subgroups.

Helmut W. Wielandt [11] called a subgroup H of G intravariant
in G if and only if for each automorphism a in A(G) there exists
a g in G such that (Ha)g = H. We will generalize this definition
to come up with

Definition 1.2.1: A subgroup H of a group G is A-intra-
variant in G for A a subgroup of A(G) if and only if for each
automorphism a in A there exists a g in G such that (Ha)g = H.

Some of the best known A(G)-intravariant subgroups are Sylow
subgroups and characteristic subgroups of any group G; and Hall sub-
groups, Carter subgroups, and system normalizers of solvable groups.

Proposition 1.2.2: If H S G then H is A-intravariant in
G for A a subgroup of A(G) if and only if A.S F(G;H)I(G).

2522;: Suppose that H is A-intravariant in G for A a
subgroup of A(G). Since I(G) A A(G), F(G;H)I(G) is a well defined
subgroup of A(G). If a 6 A then there exists a g in G such
that Hog = H. Hence ag E F(G;H) and a E F(G;H)I(G). Therefore
A S F(G;H)I(G).

If A S F(G;H)I(G) then each a in A may be represented as

Yng where y E F(G;H) and g E G. Since aéng)-1 = y we have that

n -1 ‘1
H“ g = HY = H. Therefore Hog = H and H is A-intravariant.

 

,M“

#V

 

 

 

If [G: NG(H)] is known to be equal to [A: A n F(G;H)],
where I(G) S A, then we shall show that H is A-intravariant for
H a subgroup of G. To do this we will need
Lemma 1.2.3: If H s G then F(G;H) n I(G) E NG(H)/Z(G).
nggf: Define a mapping ¢ between NG(H) and F(G;H) n I(G)
by (n)¢ = “n for n in the NG(H). This mapping is onto
F(G;H) n I(G), for if fix E F(G;H) n I(G) then x E NG(H). If y

g(xym _

Tl“ X

and x are two elements of NG(H) then

g(x)¢(Y)¢ for g in c. It then follows that ¢ is a homomorphism

 

of NG(H) onto F(G;H) n I(G).
The kernel of ¢ is equal to {n E NG(H)‘gn = g for each g
in G} = NG(H) n 2(a) = 2(6). Therefore NG(H)/Z(G) % F(G;H) n I(G).
Let us now assume that I(G) S A S A(G). Then we will be able
to derive
Proposition 1.2.4: _Let H S G and assume that I(G) S A S A(G).
Then H is A-intravariant if and only if [G: NG(H)] = [A: A n F(G;H)].
2523:: Assume that H is A-intravariant in G. From 1.1.2
we know that A S F(G;H)I(G). Hence A = A n F(G;H)I(G) =
I(G)(A n F(G;H)). From this we see that
|A| = |I(G)||A n r(c;H)|/|A n r(G;H) n I(G)‘ =
= |I(G)||An r(c;H>|/|r(G;H) n 1(c)|
Therefore [A: A n F(G;H)] = [G: NG(H)] since F(G;H) n I(G) E
NG(H)/Z(G).
Conversely; suppose that [G: NG(H)] = [A: A n F(G;H)]. We
will now show that A is equal to I(G)(A n F(G;H)). Since I(G) S A
we have that I(G) (A n 1‘(G;H)) S A. Note that

|A| = |G/z(c)||A n r(G;H)|/|NG(H)/z(c)| = |I(G)||A n r(G;H)|/|F(G;H) n I(G)l =

 

 

 

 

\I(G)(An r(c;n)|. Hence A = I(G)(A n r(c;u)) = A n r(c;u)1(c).
Therefore A S F(G;H)I(G) and by 1.2.2 we know that H is A-intra-
variant in G.

Proposition 1.2.5: If H is A-intravariant in G then NG(H)
is A-intravariant in G for A a subgroup of A(G).

2522;: If a e A then there exists g in G such that

Hag

= H. Hence NG(H)Olg = NG(Ha)g = NG(Hag) = NG(H) and we may con-
clude that NG(H) is A-intravariant.

An immediate corollary of 1.2.5 is

 

Corollapy 1.2.6: VAssume H S G with A a subgroup of A(G)
such that I(G) S A. If [A: A n F(G;H)] = [G: NG(H)] then NG(H)
is A-intravariant in G.

Egggfz This corollary follows immediately from 1.2.4 and 1.2.5.

In 1937 Philip Hall [6] proved some results which are closely
related to those which we have proven in this section of chapter I.
He considered the solvable group G with a Sylow system S. In
this paper he showed that the Sylow systems are A(G)-intravariant
in the sense that if a E A(G) then there exists a g in G such
that SOlg = S. Assume that F(G;S) is equal to the set of auto-
morphism of G which fix S. Using this notation he showed that
A(G) = I(G)F(G;S). With this he was able to arrive at an upper
bound for the order of the automorphism group of G. Later on, we
shall be able to arrive at an upper bound for the order of the auto-
morphism group of a group G. Before we will be able to do this,

we will need more basic information.

1.3 A-closed Subgroups.

A new concept which will be useful is given in

 

 

 

 

 

 

 

Definition 1.3.1: A subgroup H of a group G is A-closed

in G for A a subgroup of A(G) if:
i) H is A-intravariant in G, and
ii) H is weakly closed in NG(H).

Recall that H is weakly closed in the subgroup K if
H8 S K S G implies that H8 = H for g in G. Sylow subgroups
of a group G, and Hall and Carter subgroups of solvable groups are
examples of A—closed groups, for A any subgroup of A(G).

The first thing which we will do in this section is to arrive
at conditions which will insure us that the set of NG(H)-automorphisms
will fix each conjugate of H as a set. Then we will derive some
basic results about A-closed subgroups.

Proposition 1.3.2: If H is an A-closed subgroup of G for
A a subgroup of A(G) then H8 is A—closed for each g in G.

3522:: If a E A then (Hg)a = (Ha)ga = nga for some x
in G. This follows from the fact that H is A-intravariant in G.

If Hx S NG(Hg) = NG(H)g then ng S NG(H). Therefore

an =H and Hx =Hg and Hg is A-closed in G.

M

Proposition 1.3.3: If H S G then A(G;H) S n F(G;Hg)
géG
A(G;NG(H)).

Proof: ASSume that a E A(G;H). It then follows that
(hg)-1(hg)a E K(G;A(G;H)) for hg in H8. By 1.1.3 the multiplier
subgroup of A(G;H) is a normal subgroup of G. By the definition
of A(G;H) it is contained in H. Therefore K(G;A(G;H)) S Hg
and (hg)°' e Hg.

Suppose that a E n F(G;Hg). If g E G and h E H then

- gEG g 1
hg = (xg )a for some x in H because 0 maps H

onto

 

 

 

 

 

10

g-1 -1 ( a)—l -l
H . 0n the other hand (x8 )a = (xa) g = hg . Hence
-1 a
C!

x = hg g which is an element of H since xa E H. Therefore

-1
g gUeNGm) and ozEA(G;NG(H))-
CorollaEy 1.3.4: If H is a self-normalizing subgroup of
c then A(G;H) = n News).
86G
Proof: Apply 1.3.3.
Scott [10, Theorem 3.3.4] has shown that the core H of a

G

subgroup H of a group G is equal to the n Hg, and that it is
86G
also the maximum normal subgroup of G contained in H.

 

If NG(H) =u and HG=<1> then n News) =—:<1>. This
86G
follows from the fact that the multiplier subgroup of A(G;H) is a

normal subgroup of G. Since HG = <1>, g-lga = 1 for all g in
c and a in A(G;H). Since A(G;H) s<1> the n Hams) ‘-‘<l>.

Assume n F(G;Hg) E <1> and that NG(H) 26:. Then each x
in HG inducesggg automorphism fix in A(G;H) because [g,x] 6 HG
for each g in G. Since A(G;H) = n F(G;Hg) E <1>, [g,x] = 1
for each g in G. Therefore x E 2:33 and HG S Z(G). This gives
us a little information as to when a NG(H)-automorphism fixes each
conjugate of H in G. We will now answer the following question,
"If H < NG(H) does a NG(H)-automorphism fix each conjugate of H
in G?"

Lemma 1.3.5: If H S G then 1"(G;H) S I‘(G;NG(H)).

2529:: Suppose that a E F(G;H), n 6 NG(H), and that h E H.
In this case hnor = (n'1)°’hn°’ = (n-lha-ln)a = ((hd- )“)°’ 6 H. There-
fore 1"(G;H) s 1"(G;NG(H))-

Lemma 1.3.6: If H is F(G;NG(H))-closed for H a subgroup

of c then .1‘(c;u) =1"(G;NG(H))-

 

 

 

 

 

11

Egggf: If y E F(G;NG(H)) then there exists a g in G
such that HY = H8 and H8 s NGai)Y = NG(H). Hence H8 = HY = H.
Therefore 1y E F(G;H) and by 1.3.5 we have that F(G;H) = F(G;NG(H)).

Lemma 1.3.7: If H is a F(G;NG(H))-closed subgroup of G
then Hg is a F(G;NG(H)g)-closed subgroup of G for g in G.

3322:: If y E F(G;NG(H)g) then y E 1"(G;NG(H))fig by 1.1.7.
Hence -y = ens where a E F(G;NG(H)). Now then, (Hg)°‘ln‘g = (Ha)g = ng 1
for some x in G since H is F(G;NG(H))-closed in G. Therefore
Hg is F(G;NG(H)g)-intravariant in G.

1 If Hgy s NCO-l)g for y in G then Hgyg-l s NG(H) and

Hgyg- = H. Therefore Hgy = H8 and we have that Hg is a
F(G;NG(H)g)-closed subgroup of G.

Theorem 1.3.8:: If H is a F(G;NG(H))-closed subgroup of
G then A(G;NG(H_)) = nG 1"(G;Hg).

113323: By 1.3.? n mung) sA(G;NG(H)) s n 1"(G;NG(H)8).
By 1.3.6 and 1.3.7 we havzeihat r(c;ug) = r‘(G;NG(H)8) for each g

in G. Therefore n 1"(G;Hg) = n 1"(G;NG(H)g) and
86G 86G

A(G;NG(H)) = n I‘(G;Hg).
SEC
The converse of 1.3.8 is false. Let G be the alternating

group of degree 5. It is well known that G is a simple group.

Let H be equal to <(1,2)(3,4)>. Since G is simple NG(H)G = <1).
Note that NG(H) = <(1),(1,2)(3,4), (1,3)(2,4), (l,4)(2,3)>. Since
NG(H)G = <1> we have that A(G;NG(H)) 5 <1> because the multiplier
subgroup of NG(H) is a subgroup of NG(H)G. By 1.3.3

n 1"(G;Hg) sA(G;NG(H)). Therefore n 1"(G;Hg) =A(G;NG(H)). Since
86G gGG

H is conjugate to <(1,3)(2,4)>, we have that H is not weakly

closed in its normalizer in G. Therefore H is not F(G;NG(H))-

 

 

 

 

 

 

 

 

12

closed in G. Hence the above assertion is correct.

Observe that if H is an A-closed subgroup of G then NG(H)
is self-normalizing. In fact we have

Proposition 1.3.9: If H s G which is weakly closed in its
normalizer in G then NG(H) is self-normalizing.

2522:: If x is an element of the normalizer of NG(H)
then Hx s NG(H) and Hx = H. Hence x E NG(H). Since the nor-
malizer of NG(H) contains NG(H) we have equality.

In chapter III we will use 1.3.8 to establish the existence

 

of nontrivial normal supplements. We have found a few relationships
between 0 F(G;Hg) and A(G;NG(H)). We will now look at F(G;H)
and A(G;§:%H)) for H a subgroup of G.

Lemma 1.3.10: If H s G such that F(G;H) s A(GgNG(H))

then NG(H) A G.

2522:: If x E NG(H) then n _1 is in A(G;NG(H)) since
it is in F(G;H). Hence g-lgx = [:,x-1] E NG(H) for each g
in G. It then follows that NG(H) A G.
The converse of 1.3.10 is not true. Let G = <a,b{a9 = b3 = 1,

ba = a7b> with H = <b|b3 = I>. Let a be the automorphism of G
defined by ad = a5 and by = b. It is obvious that a E F(G;H).
It may be verified directly that a is not an element of A(G;NG(H)).
Since NG(H) is a maximal subgroup of a 3-group it is normal in G.
Therefore our assertion is true.

Using 1.3.10 we will be able to arrive at necessary and
sufficient conditions for a subgroup H of G to be a characteristic

subgroup of G.

 

 

13

Proposition 1.3.11: If H is F(G;NG(H))-closed in G
then H is a characteristic Subgroup of G if and only if
1"(G;H) = A(G;NG(H)).
nggf: Suppose H is a characteristic subgroup of G.
Then F(G;H) = A(G) and NG(H) = G, so A(G;NG(H)) = A(G) = F(G;H).
Conversely; suppose that F(G;H) = A(G;NG(H)). Using 1.3.10
we have that NG(H) A G. By 1.3.9 NG(H) = G. Hence F(G;H) =
A(G;NG(H)) = A(G). Therefore H is a characteristic subgroup of G.

Corollary 1.3.12: Gis anilpotent group if and only if

 

F(G;P) = A(G;NG(P)) for each Sylow p-subgroup of G and each prime
p dividing |c|.

3322:: The "if" part follows from 1.3.11 and the fact that if
P is a Sylow p-subgroup of G then P is F(G;P)-closed in A.

The "only if" part follows from the fact that if G is a nil-
potent group then G is the direct product of its Sylow subgroups.

The following diagram will help us keep in mind the location
of the various subgroups such as F(G;H), F(G;NG(H)), A1 = A(G;H),
A2 = A(G;NG(H)), K1 = K(G;A1), and K2 = K(G;A2) for H a Sub-
group of G. Recall that 1.1.5 says that F(G;H) s F(G;K1). By

definition we have that A(G;H) s A(G;NG(H)), hence K

G A(G)
F(G;NG(H))
A2 F(G;H)

1 S K2 0 HG.

      

 

Figure 1.

 

 

 

 

 

14

By 1.3.5 we know that F(G;H) s F(G;NG(H)). If H is
F(G;NG(H))-closed in G for H a subgroup of G then
F(G;H) = F(G;NG(H)). We can also show that F(G;H) is self-
normalizing.

Proposition 1.3.13: If H s G which is weakly closed in
its normalizer then F(G;NG(H)) is a self-normalizing subgroup
of A(G).

2532:: By 1.3.9 we know that NG(H) is self-normalizing.
Since F(G;NG(H)) is always contained in its normalizer we only
have to show that the normalizer of F(G;NG(H)) is contained in

F(G;NG(h)). If a (F(G;NG(H)) then r(c;NG(u))“ =

E NA(G)
F(G;NG(H)). If n e NG(H) and x e H then x n = y for y in
n: a- n a my a
H. Hence y=x = ((x )) =x . Therefore NG(H) sNG(H).
. a _ - a =

Since ‘NG(H) | |NG(H)| we have that NG(H) NG(H). It then
follows that F(G;NG(H)) is a self-normalizing subgroup of A(G).

Proposition 1.3.14:_ If H is F(G;NG(H))-closed in G then
F(GSH)._iS a self-normalizing Subgroup of A(G).

Proof: F(G;NG(H)) is self-normalizing by 1.3.13. By 1.3.6
F(G;H) = F(G;NG(H)). Therefore the proof of 1.3.14 is complete.

If H is F(G;NG(H))-closed in G then figure 1 on page 13
reduces to the following. Recall that K2 = K(G;A(G;NG(H)) and
K1 = K(G;A(G;H))

G A(G)

N (H) D
L F(G;H) = F(G;NG(H))

‘ A(G;NG(H))

  

' A(G;H)

 

 

 

 

 

 

15

1.4 F(G;NG(H))-closed and Eguivalences.

In this section we give some conditions which are equivalent
to F(G;NG(H))-closure of a subgroup H in G.

Proposition 1.4.1: If H is a characteristic subgroup of a
Sylow p-subgroup P1 of G then the following are all equivalent:
1) H is weakly closed in P1.
ii) H is F(G;NG(H))-closed in G.
iii) H is normal in every Sylow p-subgroup of G which contains it.

Proof: i) is equivalent to iii) follows from Theorem A in

 

the Appendix.

Assume that H is F(G;NG(H))-closed in G. If Hx s P1 for

x in G then Hx s NG(H) . Hence H = Hx, and H is weakly-closed
in P1.

Conversely; suppose that H is weakly closed in P1. If
a E F(G;NG(H)), then there exists a g in G such that Pig = P1-
Since H is a characteristic subgroup of P1, H18 = H. Therefore
H is F(G;NG(H))-intravariant in G.

Let x be in G and assume that Hx s NG(H). Since H is

a characteristic subgroup of P , we have that PI s NG(H). Since

1
Hx is a p-group in NG(H) there exists a Sylow p-subgroup P2 of
NG(H) such that Hx s P2. Hence there exists a y in NG(H) such
-1
that Hx 5 P2 = Pi. Then ny 5 P1 s NG(H). Since H is weakly
-1
closed in P1 we have that ny = H. y is in NG(H) so Hx = H

and H is weakly closed in its normalizer.
A finite group G is said to be p-normal if and only if
Z(H) s K implies that Z(H) = Z(K) for H and K Sylow p-sub-

groups of G. As an immediate corollary of 1.4.1 we have

 

 

'. 'lilllllrlllltl I I“

 

 

 

16

Corollary 1.4.2: G is p-normal if and only if Z(P) is
F(G;NG(Z(P)))-closed in G for P a Sylow p-subgroup of G.

Egggf: Take H to be equal to Z(P) in 1.4.1.

From 1.4.2 we get the Second Theorem of Gran.

Corollary 1.4.3: If Z(P) is F(G;NG(Z(P)))-closed in G for
P a Sylow p-subgroup of G then the greatest Abelian p-group which
is a factor group of G is isomorphic to that for NG(Z(P)).

nggf: Apply 1.4.2 and Theorem B in the Appendix.

If G is a solvable group then 1.4.1 can be extended to
characteristic subgroups of Hall subgroups of G.

Proposition 1.4.4: If H is a characteristic subgroup of a
Hall subgroup M in the solvable group G then the following are
all equivalent:
1) H is weakly closed in M.
ii) H is F(G;NG(H))-closed in G.
iii) H is normal in every Hall subgroup of G whose order is the

same as M and contains H.

3322:: 1) implies iii). Assume that H s M1 where M1 is a
Hall subgroup of G whose order is the same as M. Since G is
solvable, by Theorem C in the Appendix, there exists a y in G such
tha: My = M1. Hence Mi-l = M and Hy“1 s M. By i) we have that

y

H = H and Hy = H. For m in M there exists an m in M

1 1

m Y
such that m1 = tny and H 1 = (Hy)m = Hmy = Hy = H since Hm s M.

Therefore H is normal in M1.
iii) implies 1). Suppose that Hg 3 M for some g in G.
-1 -1
Hence H A M8 by iii). Now H A M so that <M, M8 > s NG(H).

G is a solvable group which contains NG(H), hence NG(H) is

 

 

 

 

 

17

solvable. By Theorem C in the Appendix there exists a y in NG(H)
-1
such that M = Mg y. Since H is a characteristic subgroup of M,

—1
= H. It then follows that Hg = H and H = Hg. Therefore

-1
Hg Y
H is weakly closed in M.

1) implies ii). If a E F(G;NG(H)) then there exists a g
in G such that My = M8. H is a characteristic subgroup of M

implies that Ha = Hg.

Therefore H is F(G;NG(H))-intravariant in G.
Note that M s NG(H). Assume that x E G and that Hx s NG(H).

Since NG(H) is solvable there exists, by Theorem C in the Appendix,

 

a Hall subgroup M in NG(H) such that Hx s M and My = M for

1

Y

1

some y in NG(H). Then Hx s M1 s M? and Hx
-1
is weakly closed in M, H = ny

s M. Since H
and Hx = H because y E NG(H).
Therefore H is F(G;NG(H))-closed in G.

ii) implies i). Assume that Hx s M for x in c. M s NG(H)

so Hx S NG(H). Therefore H = Hx and H is weakly closed in M.

1.5 An Example.

We will now illustrate these results with an example. To do

this we will need a proposition after which we will look at the example.

Proposition 1.5.1: Let G be a group generated by
{ai‘i = 1,...,n}. If T = {a E A(G)‘a:1a: E H, H A G, i = 1,...,n}
1
then T = A(G;H).

Proof: If a and Y are in T then anlawY = a-laah1 =

-1
= d d h ' .
a ah2 1 h2h1 for a, a generator of G an h1 an 2 in H
Hence T s A(G) and by definition we have that A(G;H) s T.
We will show that T s A(G;H) by induction on the number of
letters in any reduced word of G consisting of the generators

a .,an. Note that if a is some a1 and m is an integer then

1,..

 

 

 

18

t m
-m - - ..
a (am)a = a m(a0!)m = a m(ah)m E H since H A G. Let g = n aij lJ
j=1

where each aij 6 {aili = 1,...,n]. If a E T then

-m -m m m
-1 a _ it i1 i1 it a _
g g ait ... ai1 (ai1 ... ait )

a-mit(a-mit-1 a'mi 1) (amil amit- 1):): (amit)a
it it-l i1 i1 it-l it

is in H. This follows from H being normal in G and from the in-

duction hypothesis. Therefore A(G;H) s T and the proof is complete.

_ 3 2
Example 1: Let G be equal to 83 X K where 53 - <a,b a =b =1

b
and a = a2> and K = <c‘c3 = 1>.

 

Note that [Cl = 2'32 and that 33 contains 3 Sylow 2-Sub-
groups which are <b>, <ab>, and <a2b>. Hence the NGG<b>) < G.
Let H be equal to <b>. Hence lNG(H)| = 6 or 2. Since K s NG(H)
we have that ‘NG(H)‘ = 6. By 1.5.1 the automorphism a mapping a
onto a, b onto b, and c onto c2 is a NG(H)-automorphism. It
is easy to see that a fixes H and all of its conjugates. Since
H is F(G;NG(H))-closed in G, 1.3.8 tells us we should have expected
this to be the case.

It is easily seen that a fixes no proper subgroup of H
containing HG. We shall show in chapter 3 that this will help to
insure us of a group G* which is normal in G, and such that

* *
G = G H where H n G = HG.

 

c
.-

,o-v‘v "T.

 

 

 

CHAPTER II

GROUPS WHOSE SUBGROUPS ARE A-CIDSED 0R A-INTRAVARIANT

2.1 Subgroups which are Weakly Closed in Their Normalizers or
FSGQHZ-closed in G.

 

In chapter I the concept of a subgroup of a group being A-
closed, for A a subgroup of the automorphism group was discussed.
We now pose the following questions: "If each Subgroup K of a
group G is A-closed in G, what can be said about the group G?
Can some information about G be found if only certain subgroups
of G are known to be A-closed for particular choices of A?” We
will answer these questions in this chapter. Since the definition
of being A-closed is in two parts we will first examine the notion
of a subgroup being weakly closed in its normalizer.

It can be easily shown that if H is weakly closed in its
normalizer in G and H s M then H is weakly closed in its
normalizer in M. We also know that a group G is equal to the
join of its Sylow subgroups. If the group G is solvable then
Philip Hall has shown that G is the join of its system normalizers
and each system normalizer is a nilpotent subgroup of G. These
results are given in Theorems D and E in the Appendix. Recall the
fact that in a nilpotent group G, each subgroup K is subnormal
in G. That is, in the sense of Wielandt, there exists a chain of

subgroups K1,K such that K = K1 A K2 A ... A K = G. It is

2,... n

for this reason that we look at

19

 

 

 

 

 

20

Proposition 2.1.1: If each subnormal subgroup K of the sub-
group H of G is weakly closed in NH(K) then each K A NG(H).

Proof: Since K is a subnormal subgroup of H there exists
a series K = K1 A K2 A ... A Kn = H where Ki A Ki+1 for

i = l,...,n-l. We will prove by induction on the length of the chain

 

that K A NG(H). If n = 1 then K = H and it is normal in NG(H).

Assume that the proposition is true for any chain whose length
is less than or equal to m. If n = m+l then the induction hypothesis
assures us that K2 is a normal subgroup of NG(H). If x E NG(H)
then R? s K2 s NH(K1). Hence K: = K1 and K.é NG(H).

Using 2.1.1 we are now able to derive the following

Corollary 2.1.2: Let H be a solvable subgroup of G such
that each subnormal subgroup of H is weakly closed in its normalizer
in H. Then H is supersolvable.

2522;: Since H is a finite solvable group there exists a
composition series <l> = K1 A K2 A...A K.n = H such that each factor
group Ki/Ki—l where i = 2,...,n is a cyclic group of prime order.

Using 2.1.1 we have that this composition series is a normal
series of H. Since Ki/Ki-l is a cyclic group for each i, H is
supersolvable.

Corollary 2.1.3: Let H be a nilpotent subgroup of G such
that each subgroup is weakly closed in its normalizer in H. Then H
is either an Abelian group or a Hamiltonian group.

2532;: Since H is nilpotent, each subgroup of H is sub-
normal in C. It then follows from 2.1.1 that each subgroup of H is
normal in H. If H is an Abelian grOup then the proof is complete.
If H is not an Abelian group, then by definition H is a Hamiltonian

group.

 

 

21

Keeping in mind that our original question dealt with subgroups

H being A-closed in G for subgroups A of A(G) we arrive at
Proposition 2.1.4: Let H be a nilpotent subgroup of G such

that each subgroup K of H is F(G;H)-closed in G, then F(G;H) s F(G;K).
nggf: As was noted earlier, the fact that each subgroup K of

H is weakly closed in NG(K) implies that K is weakly closed in

NH(K).
Assume that K is a subgroup of H. By 2.1.1 K A.NG(H)° If

v e 1"(G;H) then KY = Kg s H. Since H s NG(K), Kg s NG(K) and

 

hence K3 = K. Therefore Ky = K, v E F(G;K), and the proof is
complete.

Note that H could have been replaced by NG(H) in 2.1.4.

Using 2.1.1, we see that if each subgroup of a Sylow p-subgroup
H of a group G is weakly closed in its normalizer then each subgroup
of H is normal in NG(H). Before we go on, recall that a group G
has a normal p-complement if it has normal subgroup of index a power
of p and of order relatively prime to p. This brings us to

Proposition 2.1.5: If each subgroup of a Sylow p-subgroup H
of a group G is weakly closed in its normalizer in H, where p is
the smallest prime dividing the order of G, then G has a normal
p-complement.

Pppgf: By 2.1.1 we see that each subgroup K of H is normal
in NG(H). By Theorem F in the Appendix we have that G has a normal
p-complement.

We will now consider the case where each subgroup of a Sylow
p-subgroup S is weakly closed in its normalizer in S, for each

prime p dividing the order of G. When this situation held for a

 

 

 

Sylow p-Subgroup belonging to the smallest prime divisor of |Gl we

were able to deduce that G had a normal p-complement. We will use
this for the basis of an induction proof. Before we may proceed we
need the following concept:

n1 n
Definition 2.1.6: Let G be a group such that ‘G‘ = p1 ...prr
where p1,...,pr are distinct primes. Let C: be an ordering of the
set of primes dividing the order of G. Assume that p1 (2 p2 c...c: pr.

Then the group G is called a Sylow tower group for the ordering <2

<...N <

if and only if G possesses a normal series <l> = Nr < Nr-l 1

n
pkk for k = 1,...,r.
Proposition 2.1.7: Let G be a group where each subgroup of a

N0 = G Such that [Nk-l: Nk] =
Sylow p-subgroup S is weakly closed in its normalizer in S for each
prime p dividing the order of G. If ‘G‘ = p21...p:r with
p1 C...C pr for c, the natural ordering of the primes then G has
a Sylow tower for this ordering of the primes.

2529;: We will prove 2.1.7 by induction on the order of G.
By 2.1.5 we know that since p1 is the smallest prime dividing ‘6‘
that there exists an N1 normal in G such that [NozNI] = p21.

Now N1 < G and satisfies the hypothesis of our proposition. Since

n n
_ 2 r
the order of N1 p2 ...pr with p2 c:p3 c:...c:pr we have that N1

is a Sylow tower group for this ordering of the primes. Hence we have

< N such that

a normal serie: of N1, say- <l> = Nr < Nr-l <... 1
[Nlclz Nk] = pkk for k = 2,3,...,r. By Theorem G in the Appendix

N2,...,Nr are all characteristic subgroups of N Since each Ni

1.
is a characteristic subgroup of the normal subgroup N1 we have that

N1 is normal in G for i = 2,...,r. Therefore G is a Sylow tower

group for the given ordering of the primes.

 

 

 

23

83; the symmetric group on 3 letters shows that without some
additional restrictions we will not be able to proceed any further.

Note that each Sylow subgroup of S satisfies the hypothesis of

3
2.1.7. It is at this point that we shall add one condition to the
hypothesis. G'(p) denotes the smallest normal subgroup of G such
that G/G'(p) is an Abelian p-group.

Proposition 2.1.8: Let G be a group where each subgroup of
a Sylow p-subgroup Gp is weakly closed in its normalizer in Gp
for each prime p dividing lGl. If Gp i G'(p) for each Such p
then G is either an Abelian group or a Hamiltonian group.

Ppggf: By 2.1.1 every subgroup of a Sylow p-subgroup S of
G is normal in NG(S). This is true for each prime dividing ‘Gl.
By Theorem H in the Appendix we have that G has a normal p-comple-
ment for each prime dividing the order of G. Therefore G is a
nilpotent group.

Using 2.1.3 we get that each Sylow subgroup of G is an
Abelian group or a Hamiltonian group. Therefore G is either an
Abelian or a Hamiltonian group since it is a nilpotent group.

Before we make any more progress we must stop and consider
the case when each H in a p-group G is F(G;NG(H))-closed in C.
It is for this reason that we prove

Proposition 2.1.9: Let G be a p-group. Then each H in
G is F(G;NG(H))-closed in G if and only if G is cyclic.

Eggpf: If G is cyclic then each subgroup of G is a char-
acteristic subgroup of G. It then follows that F(G;NG(H)) = A(G)

for each H in G. Therefore H is F(G;NG(H))—closed in G.

 

 

24

We will now prove that each subgroup H of G is F(G;NG(H))-
closed in G implies that G is cyclic. First note that 1.3.9
implies that if H s G then NG(H) is a self-normalizing subgroup
of G. Since G is a p-group, NG(H) is properly contained in its
normalizer or is equal to G. Hence G = NG(H) and H A G. H is
F(G;NG(H))-closed in G so 1.3.6 applies, and we have that
F(G;H) = F(G;NG(H)) = A(G). It then follows that each H in G is
a characteristic subgroup of G. Therefore G is either an Abelian
group or a Hamiltonian group.

If G is a Hamiltonian group then by Theorem I in the Appendix
G is the direct product of a quaternion group with an Abelian group
in which every element is of finite odd order and an Abelian group
of exponent two. Since G is a Hamiltonian p-group, G is the
direct product of a quaternion group Q and an elementary Abelian
2-group. There exists a subgroup in Q which is not characteristic
in Q. Since an automorphism of Q can be extended to an auto-
morphism of G, we have a subgroup of G which is not characteristic
in G. This is a contradiction, therefore G is not a Hamiltonian
group.

Since G is an Abelian p-group, G is the direct product of
cyclic subgroups of p power order. If there exists two cyclic
summands of the same order then there is an automorphism which per-
mutes these two summands. Hence there exists two subgroups which
are not characteristic in G. This is a contradiction. Therefore
all the cyclic summands have different orders. If there is more
than one cyclic summand in the direct product decomposition of G

then there is an automorphism of G which moves the cyclic summand

 

I
II

 

of highest order. We have once again arrived at a contradiction.
Therefore G has only onecyclic summand and is itself cyclic.

Combining 2.1.8 and 2.1.9 we are able to derive the final
theorem of this section.

Theorem 2.1.10: Let G be a group. G is a cyclic group if
and only if each subgroup of a Sylow p-subgroup Hp is F(G;Hp)-closed
in G and Hp S G'(p) for each prime p dividing lG‘.

Egggf: Assume that G is cyclic. Since every subgroup of
G is a characteristic Subgroup of G we have that each subgroup K
of a Sylow p-subgroup HP of G is F(G;Hp)-closed in G. It is
easily seen that HP i G'(p).

We will now prove the converse. By 2.1.8 G is either an
Abelian group or a Hamiltonian group. In either case G is a direct
product of its Sylow subgroups. By Theorem J in the Appendix the
automorphism group of G is the direct product of the automorphism
groups of the Sylow subgroups of G.

Let HP be the Sylow p-subgroup of G for a particular prime
p dividing |c|. Since G is nilpotent, l"(G;HP) =A(G). If K is
a subgroup of HP then by 2.1.4 F(G;Hp) S F(G;K). Hence K is
I"(HP,NH (K))-closed in HF since A(Hp) S A(G) = F(G;Hp) = T(G;K).
The above statement is true for each K in Hp. Using 2.1.9 we can
conclude that Hp is cyclic. Therefore each Sylow subgroup of G
is cyclic and hence G is cyclic.

The condition that HP S G'(p) for each prime p dividing

 

‘G‘ is a necessary condition. The symmetric grOup on 3 letters is
a group where each subgroup of a Sylow p—subgroup HP is F(SS,HP)-

closed in S3 for each prime p dividing ‘83‘, but 53 is 1

 

 

certainly not cyclic.

Wolfgang Gascthz has studied solvable groups in which every
subnormal subgroup is normal [2]. He calls such a group a t-group.
It is interesting to note that 2.1.1 implies that if each subnormal
subgroup H of G is weakly closed in its normalizer in G then
H A G. We will use this in the next section to gain some information

about A(G).

2.2 AgGQ-closed and ASGQ-intravariant Subgroups.

We will first use the existence of a subgroup H which is A(G)-
closed in G to insure us that the automorphism group of G has a
homomorphic image which is isomorphic to a subgroup of a homomorphic
image of G. The concept of a subgroup H being A(G)-closed in G
will also be useful in showing that A(G) is solvable and is a pro-
duct of two proper subgroups. It will be possible to determine the
exact order of one of the factors of the product.

In the second part of this section we will use a subgroup H
which is A(G)-intravariant to help determine if A(G) is solvable.

We have proven in 1.1.4 that A(G;H) A F(G;H) for H a sub-
group of G. If H is F(G;NG(H))-closed in G then F(G;H) =
F(G;NG(H)) and A(G;H) A F(G;NG(H)). In general we have

Proposition 2.2.1: If H S G then NA(G)(A(G;H)) = F(G;K1)
where K1 = K(G;A(G;H)).

EEESE‘ Assume that a E A(G;H) and that v E F(G;K1). Then
galg‘Y-l‘a’.Y = g_1(gY-1k)Y = kY 6 K1 since k 6 K1. Therefore F(G;K1)
is contained in the normalizer of A(G;H).

Let y be an element in the normalizer of A(G;H). If

-1 a

-l
g g is a generator of K1 then (g_1g°')Y = (gY)_1(g‘Y)IY my 6 K1-

 

 

 

TIA~ c

  

 

. I . ., .
... . _ .u‘ ..
.. .
1 a II- .o a
1 J h .n ..
. . .. . . .
.. y .. .
. . . i» . ...
.. .4 s . ..
n l
. . i
.1! t l i.
. . . e .
.. .. u IV
\ ’ .
. u n . ‘ u n
__ . < I A! e
I 1’. . w
v . u.
. ¢ .
s
. I. .. ...
' at .. ‘-
.. . .

 

 

 

 

 

 

27

Therefore v E F(G;K1) and the proof is complete.

Proposition 2.2.2: If H is A(G)-closed in G then K1 is

a characteristic subgroup of G and A(G;H) A A(G) where K1 = K(G;A(G;H)).

Proof: By 1.1.3 K1 A G and by definition K1 S HG. Since
H is A(G)-closed and HG = fl Hg we have that HC is a characteristic

Subgroup of G. Note thit iig;chl= A(G;Hi. If y E A(G) and
a E A(G;H) then gnlg'Y aY = ((gY )-1(gY )d)Y E HG for each g in
G. Therefore A(G;H) A A(G). By 2.2.1, F(G;K1) = A(G). Therefore
K1 is a characteristic subgroup of G.

At this time we will prove a well known result. To do this
we first prove

Lema 2.2.3: If H is A(G)-closed in G then A(G)/A(G;H)
is isomorphic to a subgroup of A(G/HG).

Egggfz We know by 2.2.2 that A(G;H) = A(G;HG) A A(G). Since
HC is a characteristic subgroup of G, F(G;HG) = A(G). By 1.1.6
A(G)/A(G;HG) is isomorphic to a subgroup of A(G/HG). Therefore
A(G)/A(G;H) is isomorphic to a Subgroup of A(G/HG).

If H is a characteristic subgroup of G then we have the
well-known

Theorem 2.2.4: If H is a characteristic subgroup of G
then A(G)/A(G;H) is isomorphic to a Subgroup of A(G/H).

2522:: Apply 2.2.3.

We will now consider an example to illustrate the usefulness
of 2.2.3.

Example 2: Let G be the nontrivial extension of the
quaternion group Q by the cyclic group of order 3. Then Z(G) = HG,

for H one of the cyclic subgroups of order four. Note that

 

 

 

 

 

 

 

28

Z(G) is a 2-group while the only Abelian homomorphic image of G
is a 3-group. Hence A(G;H) * <1>. If A(G;H) is not isomorphic
to <1>, there would exist a nontrivial automorphism a in A(G;H).

lga for each g in G is an endo-

The mapping of g onto g-
morphism of G onto Z(G). This is a contradiction to the above
statement. Using 2.2.3 we have that A(G) E A(G/Z(G)).

Using the properties which a subgroup H possesses when it
is A(G)-closed in G, we will give conditions which will insure us
that the automorphism group of G is solvable. In the process some
other structure theorems will be found.

Lemma 2.2.5: If H is A(G)-intravariant in G then A(G)
is solvable if and only if F(G;H)/F(G;H) n I(G) and G are solvable.

Eggpfz Assume that A(G) is solvable. It is immediate that
F(G;H)/F(G;H) n I(G) and G are solvable since F(G;H) and
I(G) are subgroups of A(G).

Conversely; suppose that G and F(G;H)/F(G;H) n I(G) are
solvable. By 1.2.2 A(G) = T(G;H)I(G). Hence A(G)/I(G) E
F(G;H)/F(G;H) n I(G). It follows that A(G)/I(G) is solvable. c
is solvable, therefore I(G) is solvable. From this we can conclude
that A(G) is solvable.

Theorem 2.2.6: Suppose that H is A(G)-closed in the solvable
group G with [G: NG(H)] a prime. If r(G;H)/F(G;H) n I(G) is
solvable with A(G;NG(H)) E <l> then there exists a minimal normal
subgroup T of A(G) such that the following are satisfied:

i) CA(G)CI) = T;
ii) T is £22 minimal normal subgroup of A(G);

iii) A(G) = F(G;H)T;

 

 

 

29

W) T n F(G;H) E <1>;
v) |T| = [G: NG(H)];
vi) if GO/T is a minimal normal subgroup of A(G)/T and
A0 = F(G;H) n Go then F(G;H) = NA(G)(AO), |GO| = |T||AO|,
‘Tl = pa, ‘AO| = qb and p # q for p and q primes.
nggg: Since H is A(G)-closed in C, it follows from 1.2.4
that [G: NG(H)] = [A(G): F(G;H)] is a prime. Hence F(G;H) is a
maximal subgroup of A(G). Using 1.1.7 and 1.3.8 we have that
A(G;NG(H)) = n mung) = n F(G;H)“ E<1>. We may conclude from
BEG aEA(G)
2.2.5 that A(G) is solvable.
Since A(G) is solvable there exists a minimal normal Sub—
group T of A(G). Hence by Theorem K in the Appendix A(G) = F(G;H)T

(T) = T,

where T is the minimal normal subgroup of G. Then CA(G)
F(G;H) n T ’é<1>, [G: NG(H)] = [A(c): F(G;H)] = |T| , and condition
vi) is satisfied.

Proposition 2.2.7: If H is A(G)-closed in G with
[6: NG(H)] a prime then A(G) = F(G;H)A(G)'.

3593;: By 1.2.4 [6: NG(H)] = [A(G): F(G;H)] is a prime.
Hence F(G;H) is anmximal subgroup of A(G). Since H is F(G;NG(H))-
closed in G, it follows from 1.3.14 that F(G;H) is a self-normaliz-
ing subgroup of A(G). Hence A(G)' is not a subgroup of F(G;H)
and A(G) = F(G;H)A(G)'.

Continuing in our search for groups whose automorphism groups
are solvable we arrive at

Proposition 2.2.8: Let each subgroup in a composition series
of the subgroup H of the solvable group G be F(G;H)-closed for H
an A(G)-intravariant subgroup of G. If F(G;H) is solvable then

A(G) is solvable.

 

 

 

30

M: By 2.2.5 A(G) will be solvable if F(G;H)/F(G;H) n I(G)
is solvable. If we are able to Show that F(G;H) is solvable then
we will be able to conclude that A(G) is solvable.

First note that F(G;H) A F(G;H). If v E F(G;H) and

-1

a€F(G;H) then h" aY=h for each h in H. Let
<1> = NrSNr_1S...S N0 = H be the composition series of H mentioned
in the hypothesis of 2.2.8. By the proof of 2.1.1 each Ni is a
normal subgroup of H. If a E F(G;H) then N: = N: s H for some
g in G. Since N? S H S NG(Ni) we have that N: = N1 for each
i. Therefore F(G;H) leaves each subgroup in the series fixed. One
can easily show that F(G;H)/F(G;H) is isomorphic to a subgroup of
A(H). By Theorem L in the Appendix F(G;H)/F(G;H) is solvable. It

is given that F(G;H) is solvable, hence F(G;H) is solvable and

the proof is complete.

2.3 An Upper Bound for the Order of Ang.
We will now derive an upper bound for the order of A(G)
using the concept of H being A(G)-intravariant in G.
R Proposition 2.3.1: Let H be an A(G)-intravariant subgroup

of G with the Hrautomorphisms of G containing F(G;K1) where

x =K(G;A(G;H)). Then |A(c)| divides LG:NG(H)]|A(1<1)||T(G;1<1)|
e

l
e
_ 1 n . . _
where ‘K1| p1 ... pn With p1,...,pn primes and e1,...,en non
negative integers. |F(G;K1)‘ = p11 ... pnn for nonnegative integers
a1,...,an.

Proof: If p is a prime dividing |F(G;Kl)‘ then there
exists an a in F(G;K1) Such that ‘a‘ = p. Since a is not the
identity, there exists a g in G such that g-lgy = k 6 K1, k # 1

P
and ga = g k. Note that gd = g kp = g. Therefore kp = 1 and

 

 

 

31

p divides |Kl|. It follows from the above that
. _ 1 n . .
\F(G,K1)| - p1 ...pn for some nonnegative integers al,...,an.
By 1.1.5 K is admissible with respect to F(G;H). Define

1

¢ mapping F(G;H) onto A(Kl) by kW)‘23 = kY for each k in K1.

Since y is an automorphism of G (y)¢ is an automorphism of K1.
The kernel of ¢ is equal to {v E F(G;H)‘kY = k for each k in K1} =
F(G;K1). Therefore F(G;H)/F(G;K1) is isomorphic to a subgroup of
A(Kl)'

It follows from 1.2.4 that [A(G): F(G;H)] = [G: NG(H)]. Since

[G: NG(H)]|A(K1)‘|F(G;K1)|.

We are now in a position to prove

Proposition 2.3.2: Let H be a characteristic subgroup of G
such that F(G;K1) S A(G;H) where K1 = K(G;A(G;H)) is a cyclic p-
group of G. Then |A(G)| divides pm-1(p-l) for some integer m.

3323;: If |x1| = 2n then by 2.3.1 \A(c)| divides
‘A(K1)|2t for some integer t. Since K is a cyclic 2-group it

1
follows from Theorem M in the Appendix that |A(K1)| = 2n-l. There-

|F(G;H)/F(G;K1)‘ divides \A(K1)‘ we have that [A(G)‘ divides
l
l
1
l
fore ‘A(G)| divides some power of 2.

Assume that the order of K1 is equal to pn for an odd
prime p. Then by Theorem M in the Appendix we have that |A(Kfl =
pn'1(p-1). By 2.3 1 |A(c)| divides pn‘1(p-1)pt for some integer
t .

”.Corollary 2.3.3: Let H be a characteristic subgroup of G
such that K1 = K(G;A(G;H)) is a cyclic p-group with p a prime of
the form 2m + l for some integer m. If F(G;K1) S A(G;H) then

[A(G)‘ divides pt2m, A(G) is solvable, and G is solvable.

 

1
\
\

I”

9

.0

 

32

Eggpf: By 2.3.2 lA(G)| divides anIn for some integer m.
By Theorem N in the Appendix A(G) is solvable. I(G) S A(G) so we
have that G/Z(G) is solvable which implies that G is solvable.

Another easy corollary of 2.3.2 is I

Corollary 2.3.4: If H is a normal Sylow 2-subgroup of G
such that F(G;K1) S A(G;H) for K1 = K(G;A(G;H)) cyclic then
G = H.

2522:: By 2.3.2 [A(G)‘ divides 2n for some n. It then
follows that A(G) is nilpotent which implies G/Z(G) is nilpotent.
Therefore G is nilpotent.

ASSume that there exists a prime p f 2 which divides ‘Gl.

A nilpotent group is a direct product of its Sylow subgroups. Hence,
by Theorem J in the Appendix, A(G) is a direct product of the auto-
morphism groups of the Sylow subgroups of G.

Let a be an element of A(Hp) where Hp is the Sylow p-
subgroup of G. It is showu in the proof of Theorem J in the Appendix
that a can be considered as an automorphism of G which acts as
the identity on H. It then follows that a E A(G;H) since
F(G;K1) S A(G;H). If g e Hp then g_1ga E H n Hp = <1>. Therefore
a is the identity automorphism and A(Hp) 5 <1>. Since 2 < p we
have a contradiction. Therefore H is the only Sylow subgroup of

G and G = H.

 

 

 

CHAPTER III

H-AUTOMORPHISMS AND N G (H) -AUTOMORPHISMS

We will start chapter III by deriving a condition which will
guarantee the existence of an H-automorphism. From this point on a
major portion of the results will be aimed at deriving conditions
which will assure us of the existence of a complement or a non-
trivial supplement of H in G. A supplement of H in G is a
subgroup K of G such that G = HK. The results of chapter I
will be used frequently to establish the existence of the above
groups. Most of the results will be based on the fact that one is
able to find a H-automorphism or a NG(H)-automorphism which acts in
a particular way on certain Subgroups of G.

In chapter I we found that if a subgroup H was F(G;NG(H))-
closed in G then each NG(H)-automorphism fixed each conjugate of
H in G. This knowledge will be useful in establishing the

existence of a nontrivial supplement of H in G in 3.3.5.

3.1 NG(H)-automorphisms which are H-automorphisms.

 

In chapter I we found a necessary condition for A(G;NG(H))
to be equal to n F(G;Hg) for H a subgroup of G. Generally
speaking, NG(H) 85211 be larger than H and hence it will be easier
to find a NG(H)-automorphism than an H-automorphism. We will now
find a necessary condition for a NG(H)-automorphism to be an H-
automorphism. Once this condition is found, information will be

33

 

 

   
   
    

71‘ * j"
h. ‘

34

gained about the group based upon the existence of a H-automorphism
or a NG(H)-automorphism.

Recall that if a 6 A(G;NG(H)) and H is F(G;NG(H))-closed
in G then H is a invariant. This leads to

Proposition 3.1.1: Let H be F(G;NG(H))-closed in G and
assume that a E A(G;NG(H)) such that (‘a‘, [NG(H)G‘) = 1. If a
acts as the identity on NG(H)G/H n NG(H)G then a E A(G;H).

Egggfz From the definition of A(G;NG(H)) we have that
ga = g f for f in NG(H)G and g in 6.2 Assume that ‘a‘ = n.
If we operate on ga by a we get that gm = gag: = g fzx for

some x in H n NG(H)G. Continuing in this manner we have that

n
ga = g = g fny for some y in H n NG(H)G. This is because

n

_ -l
H n NG(H)G A'NG(H)° It then follows that f — y e u n NG(H)G.

Since (n, ‘NG(H)G‘) = l we have that f E H n NG(H)G s H. Hence
£1 = g f where f is in H for each g in G. Therefore
a E A(G;H).

If H happens to be a Sylow subgroup of G then H is
F(G;NG(H))-closed in G and we get

Corollary 3.1.2: Let H be a Sylow p-subgroup of G with
a in A(G;NG(H)). If (M, |NG(H)G|) = 1 and 0, acts as the
identity on NG(H)G/H n NG(H)G then a e A(G;H).

Egggi: Apply 3.1.1.

The symmetric group on 4 letters shows that the converse of
3.1.2 is false. Each element in the Klein 4-group induces a Sylow
2-subgroup automorphism, but the order of the Klein 4-group is four.
It can also be shown that the Klein 4-group induces the set of all

Sylow 2-subgroup automorphisms of G.

 

 

‘I

 

3.2 H-automorphisms.
If H s G and K is the multiplier subgroup of A(G;H),
then by 1.1.3 we know that K is a normal subgroup of G which is
in HG. There are a number of easily proven results that can be found
if one knows the existence and the action of an H-automorphism.
Proposition 3.2.1: If H S G and a 6 A(G;H) such that
Fd(G) n H = <1> then H is the multiplier subgroup of A(G;H).
Egggf: Since H is a-invariant, a is a fixed-point-free
automorphism of H. By Theorem 0 in the Appendix H = {h-lhalh E H}
which is a subgroup of the multiplier subgroup of A(G;H). Since
H contains the multiplier subgroup of A(G;H) we have that they
are equal.
The converse of 3.2.1 is false. Let G be the Abelian group
of order 8 which is the direct product of a cyclic group of order 4
and a cyclic group of order 2. ASSume that G = <a|a4 = l> X
<b b2 = 1>. Let H be euqal to <a2b>. If a is the automorphism
of G given by ad = a3b and bu = b then 1.5.1 tells us that a
is a H-automorphism. By direct calculation one can show that a is
the only nontrivial H-automorphism. Note that (azb)a = (a3b)2b = azb.
Therefore the converse of 3.2.1 is false.
Using 3.2.1 we can easily deduce
Corollary 3.2.2: Suppose that H is a Hall subgroup of G.
If there exists an a in A(G;H) such that Fa(G) n H = <1> then
G has a H-complement and all H-complements are conjugate in G.
Egggfz By 3.2.1 H A G. By Theorem P in the Appendix G has

an H-complement and all H-complements are conjugate.

Another easy consequence of 3.2.1 is

 

 

 

36

Corollary 3.2.3: If for each maximal subgroup H of G
there exists an a in A(G;H) such that Fa(G) n H = <1> then
G is nilpotent.

EEEEE: By 3.2.1 H is the multiplier subgroup of A(G;H).
Hence H A G and G/H is cyclic. Therefore G' S H for each
maximal subgroup H. It then follows that G' is contained in the
Frattini subgroup of G which implies that G is nilpotent.

Using one of Wielandt's theorems, TheoreulQ in the Appendix,
and the concept of A-intravariance we are able to prove

Lemma 3.2.4: Suppose that H is a subgroup of G which is
self-normalizing in G. If there exists an a in A(G;H) such
that a fixes no proper subgroup of H containing HG then there

* * *
exists a subgroup G of G such that G A G, G = HG , and

Proof: Since a is in A(G;H) we know by 1.3.3 that a

a
8) 8

fixes each conjugate of H in G. Hence if g E G, then (H = H

and (H n Hg)“ = Hn Hg. Now then HG s an Hg s H. If g is not

in H then g is not in NG(H). Under this assumption, H n Hg < H.

Therefore H n Hg = HG for all g in G - H. Now HG _A_ G, so by

TheoreuzQ in the Appendix there exists a G* in G such that

* * *

G A G, G = G H, H n G = HG.
Proposition 3.2.5: Suppose that H is a maximal subgroup of

the group G which has all its proper subgroups solvable. If there

exists an a in A(G;H) such that a fixes no proper subgroup of

H containing HG then:

i) G is solvable, and

ii) G/H or H/HG is a cyclic group of prime order.

 

 

Proof: If H A G then G/H is a cyclic group of prime order.

Since H is solvable, G is solvable.

If H is not a normal subgroup of G then H is self-normal-
izing because H is maximal in G. By 3.2.4 there exists G* in G
such that 6* A G, G = HG*, and H n G* = HG. Since HG < H, G* < C.
If G* = G then G* n H would be equal to H. It then follows that
G/G* is solvable because G/G* E H/HG and H is a proper subgroup
of G. G* is a proper subgroup of G, so it is solvable. Therefore
G is solvable.

If x E G then x-lxa is an element of the multiplier sub-
group of A(G;H). Therefore x-lxa is in HG since the multiplier
subgroup of A(G;H) is a normal subgroup of G in H. It then
follows that a acts as the identity on G/HG and hence on H/HG.
If T/HG < H/HG then T/HG is left fixed by a. If t E T then
t°’ = t y e T because y e H s T. Since HG < H and H/HG has no
proper subgroups we can conclude that H/HG is a cyclic group of
prime order.

AS, the alternating group of degree 5 is a group which has all
of its proper subgroups solvable. If H is a maximal subgroup of
A5 then A(G;H) E <1> because A5 is a simple group. Therefore

A5 doesn't have an a as described in 3.2.5.

3.3 NG(H)-automorphisms.

We have shown so far, that if a certain H-automorphism a
could be found for a particular subgroup H then there exists a
nontrivial supplement in G. We will now switch the emphasis to

NG(H)-automorphisms.

 

 

 

38

Proposition 3.3.1: Suppose that H s G such that if NG(H) s L
then NG(L) = L. Assume there exists an automorphism a in A(G;NG(H))
such that Fa(G) n K2 = <1> where K2 = K(G;A(G;NG(H)) and that a
restricted to NG(H) is a Z(NG(H))-automorphism. Then there exists
a normal subgroup M of G Such that G = M X K2 and a E A(G;Z(G)).

Ppppfz Since a 6 A(G;NG(H)), a restricted to NG(H) is a
well defined automorphism of NG(H). By 1.1.5 K2 is a invariant.

Hence a is fixed-point-free on K and K2 = {x-lxalx 6 K2} by

2

Theorem 0 in the Appendix. Since K2 5 NG(H) we have that

K2 5 Z(NG(H)). It follows from this that NG(H) s CG(K2). By 1.1.3
K2 A G, hence CG(K2) A G. On the other hand NG(CG(K2)) = CG(K2).

Therefore G = CG(K and K2 s Z(G).

2)
Define y mapping G onto K2 by gY = 3-

shown that y is a homomorphism of G onto K2. Therefore

1g”. It is easily

G/ker y 5 K2. Let M be equal to ker y. Therefore M A G.
If g E M n K2 then g-lgy = 1, but g G K2 implies
g = g = 1. Since M = [M K2| = |M||K2| we have that G = M x K2.
Burnside has shown, Theorem R in the Appendix, that if a Sylow
subgroup of G is contained in the center of its normalizer that
G has a normal p-complement. Hence we will call a subgroup H of
a group G a B-subgroup of G if H S Z(NG(H)) implies that H
has a normal H-complement. This brings us to
Theorem 3.3.2: Suppose that H is an Abelian B-subgroup of
G which is a Hall subgroup of its normalizer in G. If M is a
complement of H in NG(H), a E A(G;NG(H)), Fa(G) n M = <1>, and

(|K(G;<a>)|,|H|) = 1 then G has a normal H—complement and

H S Z(NG(H)).

 

 

39

2322:: By Theorem P in the Appendix, NG(H) has a H-complement
M and all the H-complements in NG(H) are conjugate.
If H = NG(H) then H = Z(NG(H)) since H is an Abelian
subgroup of G. Therefore G has a normal H-complement.
If H < NG(H) then H has a nontrivial complement M in
NG(H) such that Fa(G) n M = <1>. We will now show that M = K(G;<a>).
First note that by 1.1.5 that K(G;<a>) A G. Therefore
M K(G;<a>) is a well defined subgroup of G. M s M K(G;<a>) s NG(H)
since a E A(G;NG(H)). M is a complement of H in NG(H) So
H M K(G;<a>) = NG(H)' It follows that
|M K(G;<a>)| = |M| |K(G;<a>)|/|M n K(G;<a>)|. Therefore
(lH|, |M K(G;<a>)|) = 1. Since M K(G;<a>) is a complement of H
in NG(H), there exists a y in NG(H) such that
M = (M 1<(<3;<o,>))y = MyK(G;<a>). It then follows that K(G;<cy>) s M.
If m E M, then H” = xh = mt for some x in M, t in
K(G;<a>), and h in H. Hence h = x-lmt 6 H n M = <1>. From this
we see that h = l and M is an a invariant subgroup of G.
Since Fa(G) n M = <1>, a is a fixed-point-free automorphism of M.
Therefore Theorem 0 in the Appendix tells us that
M = x-lxa|x E M} s K(G;<a>). It is clear that M = K(G;<a>) A G.
From the above we have that NG(H) = H X M where H is an
Abelian group. Therefore H s Z(NG(H)) and G has a normal H
complement.
If we let H be an Abelian Sylow p-subgroup of G we get
Corollary 3.3.3: Suppose that P is an Abelian Sylow p-Sub-
group of G. If M is a complement of P in NG(P),

a 6 A(G;NG(P)), Fa(G) n M =<1> and (|K(G;<a>)|, p) = 1 then G

 

 

40

has a normal p-complement and K(G;A(G;P)) = K(P;A(G;P)) S Z(G).

nggf: P is a B-group by Theorem R in the Appendix. It is
a Hall subgroup of its normalizer in G since it is a Sylow p-sub-
group of G. Therefore 3.3.2 implies that G has a normal p-
complement N.

Note that G = NP. Since N is a normal Hall subgroup of G,
by Theorem G in the Appendix, N is a characteristic subgroup of G.
If a 6 A(G;P) then n-lna 6 P n N = <1> for each n in N.

If g E G then g = nx for n in N and x in P. There-

-1 -1 _
0’ X n lnaxa

fore g g = -1xa

for each a in A(G;P). From this
it follows that K(G;A(G;P)) = K(P;A(G;P)). Let K1 = K(G;A(G;P)).
then K1 S P. By 3.3.2 we know that P S Z(NG(P)). Therefore
NG(P) S CG(P) S CG(K1).

Since K1 A G the cG(K1) A G, but NG(P) s cG(1<1). There-
fore NG(CG(K1)) = CG(K1) = G. It then follows that K1 S Z(G).

Example 1 in chapter I is an example of a group G which
satisfies the hypothesis of 3.3.3. If one takes P to be a Sylow
2-subgroup of G and a to be as given , then one can verify
directly that all the hypothesis are satisfied.

If H is an Abelian Hall subgroup of G then we could let
H be equal to P in 3.3.3. The conclusion would be the same as
that of 3.3.3 because Theorem S in the Appendix assures us that H
is a B-group of G.

If a Hall subgroup H of G of odd order satisfied the
hypothesis of 2.1.8 as a group then H would either be an Abelian

group or a Hamiltonian group. This leads us to

 

 

 

 

41

Corollary 3.3.4: Let H be a Hall Subgroup of G of odd
order such that each subgroup of a Sylow p-subgroup Hp is weakly
closed in its normalizer in Hp for each prime p dividing the
order of H. If Hp S GED), a E A(G;NG(H)), M is a complement of
H in NG(H), Fa(G) n M E <1>, and (|K(G;<a>)|, |H|) = 1 then

G has a normal H-complement.

 

Pppgf: By 2.1.8 we have that H is either an Abelian group
or a Hamiltonian group. By Theorem I in the Appendix H is an
Abelian group since it has odd order. By Theorem P in the Appendix
NG(H) has a normal H complement. By Theorem S in the Appendix
H is a B-group in G. We are assured that G has an H complement
by 3.3.2.

If we do not know that H is a Hall subgroup of G, then we
can still find a supplement of H in G if some knowledge of the
intersection of the conjugates of H is known. This brings us to
Theorem 3.3.5.

Theorem 3.3.5: Suppose that H is a Hall subgroup of F,
its normalizer in G. If M is a complement of H 'in F, M S FG,
and a E A(G;F) such that a fixes no proper subgroup of H con-
taining HG then there exists a unique G* in G such that
G* A G, G = G*H and HG = 6* n H.

Pppgf: First we will show that H is F(G;F)-closed in G.
If y E F(G;F) then HY = H because Theorem G in the Appendix

assures us that H is a characteristic subgroup of F. Assume that

 

H8 S F for g in G. Then H H8 S F and |H Hg‘ = |H||Hg|/|H n Hgl.

 

If d = ‘H n Hg| then d divides m where ‘F‘ = m n and ‘H‘ m.

Now ‘H Hs| = mz/d and m2/d divides m n. Since (m,n) = 1, m = d

 

42

8

and H = H . Therefore H is F(G;F)—closed in G. Since a 6 A(G;F),

1.3.8 assures us that a 6 fl F(G;Hg).
gEG
* * * *
If HAG then G=G. If G <G then H=HG=G nHSG
* *
and G = G H = G < G which is a contradiction.
Hence we may assume that H is not a normal subgroup of G.
If g E G - F then HG S H n Hg < H. By the above we know that
8 a _ 8 = 8
(H n H ) - H n H . Therefore HG H n H for each g in G - F.
By Theorem P in the Appendix F has an H complement M and
M S FG. Since F = M H we have that [F : H] = lM‘ and that
[H : HG] divides lHl. Therefore ([F : H], [H : HGJ) = 1.
Since HG A G and HG S F we have that HG S FG. Note that
H is a invariant and that HG S FG n H S H. a fixes FG n H

so FG n H = H or FG n H = H.

G
If FG G H = H then F = M H S FGH = PC. In this case F A G.

We showed earlier that H is F(G;F)-closed in G, hence it follows

from 1.3.9 that F is its own normalizer. Therefore F = G and

H A G which is a contradiction. Hence HG = FG n H.

Note that = FG n F = FG n H M = M(FG n H) = M H . Hence

FG G
[F : FG] = |M||H|/|M||HG| = [H : HG]. Using TheoremT in the

*
Appendix we are guaranteed the existence of a unique G in G such

*
that G = G H and HG

*
=G nu.

Example 1 in chapter I is again an example of a group G
which satisfies the hypothesis of 3.3.5. If H is taken to be a

Sylow 2-subgroup of G and a is used as given then one can verify

directly that the hypothesis are actually satisfied.

 

CHAPTER IV

INNER H-AUTOMDRPHISMS

In this chapter we will consider those elements of a group G
which induce H-automorphisms for H a subgroup of G. The set of
these inner H-automorphisms will be denoted by H. If some knowledge
of this set is known, one is able to find supplements and in some cases

complements of HG in G.

4.1 Basic Properties of H.
Definition 4.1.1: If H S G then we will let
H = {x E G|[g,x] E H for each g in G}.
Proposition 4.1.2: If H S G then H A G.
3522:: If x and y are elements of H then
[g,xy] = [g,y][g,x][g,x,y] which is an element of H for each g
in G. Hence H S G.
We will now show that HAG. If x6H,n€G, and gEG

then [g,xn] 5 [g,n-1x n] a g_1n—1x_1n g n_1x n s

1x-lx n—lx-ln g x x-ln-lx n i g_1x-1[x_l,n]g x[x,n] E [g,x] s

i g-
5 1 mod H. The reason for this is that [x,n], [x-1,n], and [g,x]
are all in K(G;A(G;H)) which is a normal subgroup of G. There-
fore H A C.

At this point we should note that Z(G) S H S NG(H) for H

a subgroup of G. The relationship between A(G;H) and H is a

very natural one and is given in

43

 

 

44

Proposition 4.1.3: if H S G then A(G;H) n I(G) °—‘ H/Z(G).
m: Define a mapping of H into A(G;H) n I(G) by
x¢ = "x for each x in H. Since g()<y)¢ = gx¢y¢ for x and y
in H, it follows that ¢ is a homomorphism of H into A(G;H) n I(G).
From the definition of A(G;H) we see that the mapping is onto
A(G;H) n I(G). The kernel of ¢ is equal to {x E ngx¢ = g for
all g in c} = in 2(a) = 2(a). Therefore H/z(c) '5 A(G;H) n I(G).
The relationship of H to H is given in
Proposition 4.1.4: If H S G then H n H = HG.
P_ro_o_f_: Since HG A G and HG S H, we have that HG S H n H.
If y E H n H then by 4.1.2 yx E H for each x in G. Note that
yx = y[y,x] = y[x,y‘_‘-1 is an element of H since y E 5.0 H. Hence
H n H A G which is contained in H. Therefore H n H S H and

G

HG = in H.

Corollary 4.1.5: If H = NG(H) < G then HG = H.

ngpf: Since H A G and H S NG(H) = H we have that
H S HG = H'n H S H. Therefore H = HG.

The converse of 4.1.5 is not true as AS, the alternating
group of degree 5, illustrates. Since A5 is simple, HG = H = <1>
for each proper subgroup of A5. Not all the subgroups of A5 are
self-normalizing. The normalizer of the Klein 4-group is the
alternating group of degree 4.

Proposition 4.1.6: If H S G then H is admissible with
reSpect to F(G;H).

2522:: Assume that v 6 F(G;H) and x E H. Then
[g,xY] = [gY-1,x]'Y E H since [gY‘1,x] E H. Therefore H is

admissible with respect to F(G;H).

 

 

 

45

Corollary 4.1.7: If H is A(G)-intravariant in G then H
is a characteristic subgroup of G.

Proof: By 1.2.2 we have that A(G) = F(G;H)I(G). It then

 

follows from 4.1.2 and 4.1.6 that H is a characteristic subgroup
of G.
Proposition 4.1.8: If H S G then H = H; for each y in G.
2522:: If x 6 H then [g,x] is an element of the multiplier
subgroup of A(G;H) for each g in G. Since K(G;A(G;H)) A G
which is contained in H we have that [g,x] 6 Hy for each y in

G. Therefore H S Hy. Hy S H follows by a similar argument. There-

fore H = Hy.

4.2 H and Supplements of HG in G.

 

If H can be found for a particular choice of H then we will
be able to show that the core of H has a Supplement. It is natural
to ask, "If K is a supplement of HG in G, what restrictions
placed upon HG and H will assure us that K is a complement of
HG in G?" This brings us to

Theorem 4.2.1: If P is a Sylow p-subgroup of G then P
has a PG complement M and G = PGNG(M).

P_rgo_f_: By4.1.4 PnP=PG. Since PAG,PG=PnP isa
Sylow p-subgroup of P. PC A G and hence is normal in P. By
Theorem P in the Appendix P has a p-complement M and all the
complements of PG in P are conjugate in P.

If g E G then Mg S P since P A G. Since M8 is a comple-
ment of PG in P there exists a y in P such that ng = M.

Hence gy = Z G NG(M) and g = Z y_1. It then follows that

G = NG(M)P = Ncmm PG = NG(M)PG.

 

 

46

We will now arrive at conditions which will assure us that PG
has a complement in G. First we prove

Lemma 4.2.2: Let P be a Sylow p-subgroup of G such that
PC is an Abelian group. Let M be a p-complement of P then
PG n NG(M) s 2(3).

mg; Note that PC n NG(M) A NG(M) n '13 since PC A G.
Since P = PGM we will show that each element of PG n NG(M)
commutes with each element of PG and M separately. PG is an
Abelian group, hence [P , N604) 0 PG] =<1>- EM, PG (1 BG(M)] S
s M n PC = <1>. Therefore PG n NG(M) s Z(P).

Proposition 4.2.3: Let P be a Sylow p-subgroup of G such
that PG is an Abelian group and PG 0 Z(P) =<l>. Let M be a
complement of P

G
Proof: By 4.2.1 G = PGNG(M). It follows from 4.2.2 that

' _ = d = 1>.
in P then G PGNG(M) an PGnNG(M) <

PG n NG(M) s Z(P). 0n the other hand PG n NG(M) = PG n NG(M) n Z(P) =
= <1>. Therefore PG 0 NG(M) = <1>.

If we know that |A(M)‘ and p are relatively prime and G
is a solvable group we arrive at

Proposition 4.2.4: Let G be a solvable group with Sylow
p-subgroup P. If M is a p-complement in P and (|A(M)|, p) = 1
then G = PGCG(M) and MS Z(H) for some Hall subgroup H of G
containing M where |P| = [G : H].

3333:: By 4.2.1 G = PGNG(M) where M is a p-complement in
P. Since G is solvable, by Theoremll in the Appendix, there exists
,P of

1,... n

By Theorem C

a complete set of pairwise commuting Sylow subgroups P

G such that; G = P1...Pn. Assume that |P| = IP1"

in the Appendix MS (P2---Pn)x for some x in G. Let H be

 

 

47

equal to (P .Pn)x. Hence M S H and G = Px H. Since P is a

2" 1
Sylow p-Subgroup of G there exists a g in G such that P? = P8.

Since M S P, [H, M] S H n P = <1>. It then

By 4.1.8 3 = pg = ;§.
follows that M S Z(H) and H S CG(M). Therefore ‘NGGM)/CG(M)l
divides some power of p. On the other hand NG(M)/CG(M) is iso-
morphic to a subgroup of the automorphism group of M whose order
'is relatively prime to_ p. [Thereforer NG(M) = CG(M) and G = PGCG(M).
We will now derive some easy consequences of 4.2.1.
Proposition 4.2.5; ’If P is a Sylow p-subgroup of G for
the smallest prime p dividing |Gl and PG is cyclic then
G = NG(M) where M is a complement of PG in P.

Proof: It follows from 4.2.1 that P = PGM where M is

a complement of PG. By Theorem M in the Appendix ‘P/CP(PG)l‘pn-1(p-l)
since PC is a cyclic p-group. Since p is the smallest prime
l l - = 1.
dividing the order of c and PG s C§(PG) s PGM, lP/GF(PG)‘
Hence P = GP(PG) and M S GP(PG)' Therefore P = PG X M. Now M
is a normal Hall subgroup of P and is therefore characteristic
in '5 by Theorem 6 in the Appendix. Since '1; A G, M A G and
NG(M) = C.
Using 4.2.1 we deduce the existence of a normal supplement.
Proposition 4.2.6: Suppose that P is a Sylow p-subgroup of
G and M is complement of PG in P. If NG(M)/PG n NG(M) has
a normal p-complement then G has a normal p—supplement S such
that S n P = PG and conversely.
- . . = . E P E
Proof. By 4 2 l G NG(M)PG Hence G/Pb PGNG(M)/ G

E NG(M)/PG n NG(M). Hence G/PG has a normal p-complement S/PG.

 

 

 

S.
K.“

 

48

Therefore G/PG = P/PGS/PG and P/PG n S/PG ! <1>. From this
statement we have that G = P S where P n S = PG.

Conversely; suppose that S is a normal p-supplement of P
in G such that P n s = PG. By 4.2.1 G/PG °-‘ NG(M)/PG n NG(M).
Since G/PG '-‘—- (S/PG)(P/PG) where s n P = P , NG(M)/PG n NG(M)
has a normal p-complement.

I have not been able to find any examples where M < NCO/I) < G
for M a p-complement of PG in P. ‘If one looks at S3, we see
that if we take P to be the Sylow 3-psubgroup that P = S3 = P M
where M is a self normalizing Sylow 2-subgroup. If we take M to
be a Sylow 2-subgroup of 83, then P =<l> and NS3(M) = $3. It

would be interesting to know if there exists an example of a group

G such that M< NG(M) < G.

 

 

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BIBLIOGRAPHY

 

10.

11.

 

BIBLIOGRAPHY

C.W. Curtis and I. Reiner, Representation Theory of Finite

Groups and Associative Algebras (New York: John Wiley and
Sons, Inc., 1962).

W. Gaschutz, Gruppen in denen das Normalteilersein Transitiv

ist. Journal fur die Reine and Angewandte Mathematik, 198
(1957), 87-92.

D. Gorenstein, Finite Groups (New York; Evanston, and London:
Harper and Row, Publishers, 1968).

 

M. Hall, The Theory of Groups (New York: The Macmillian
Company, 1964).

P. Hall, A note on Solvable Groups, Journal London Mathematical
Society, 3 (1928), 98-105.

, 0n the Sylow Systems of a Solvable Group, Proceedings
London Mathematical Society, 43 (1937), 316-323.

, System Normalizers of a Solvable Group. Proceedings
London Mathematical Society, 43 (1937), 507-528.

H. Liebeck, The Automorphism Group of Finite p-groups, Journal
of Algebra, 4 (1966), 426-432.

D.J.S. Robinson, A Note on Finite Groups in which Normality is

Transitive. Proceedings of the American Mathematical Society,
19 (1968), 933-937.

W.R. Scott, Group Theory (Englewoods Cliffs, New Jersey:
Prentice Hall, Inc., 1964).

H. Wielandt, Topics in the Theory of Composite GrOups «disconsin:

Department of Mathematics, University of Wisconsin, 1967).

49

 

 

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APPENDIX

 

 

APPENDIX

I _Theorems

Theorem A I3]: Let K be characteristic subgroup of the Sp-
subgroup P of G. Then K is weakly closed in P if and only if
K is normal in every SP-subgroup of G in which it lies.

Theorem B [4]: (Gran) If G is p-normal, then the greatest

 

Abelian p-group which is a factor group of G is isomorphic to that
for the normalizer of the center of a Sylow p-subgroup.

Theorem C [5]: (P. Hall) Let G be a solvable group of order
mn where (m,n) = 1. Then
1) G possesses at least one subgroup of order m.
2) Any two subgroups of order m are conjugate.
3) Any subgroup whose order m' divides m is contained in a sub-

group of order m.

Theorem D [7]: (P. Hall) The join of all the system normalizers
of a solvable group G is G itself.

Theorem E I7|: (P. Hall) A system normalizer is the direct
product of its Sylow subgroups.

Theorem F [9]: (D.J.S. Robinson) If the finite group G
satisfies CP where p is the smallest prime dividing the order of
G, then G has a normal p-complement. A group G is said to satisfy
the condition cP if every subgroup of a Sylow p-subgroup S of G is

normal in NG(S)-

50

 

 

51

Theorem G ]3]: If H is a normal subgroup of G whose order
and index are relatively prime, then H is characteristic in G.
Theorem H ]9]: (D.J.S. Robinson) If the finite group G

satisfies Cp, G either has a normal p-complement or G S G' for

p (p)
Gp a Sylow p-subgroup of G.

Theorem I ]4]: A Hamiltonian group is the direct product of
a quaternion Q = <a,b]a4 = 1, a2 = b2 = 1, ab = b a—1> with an
Abelian group in which every element is of finite odd order and an
Abelian group of exponent two.

Theorem J ]10]: If G is the direct product of the char-
acteristic subgroups H1,...,Hn then the A(G) is the direct product
of A(Hl),...,A(Hn).

Theorem K ]11]: Let G be a finite solvable group, A a
maximal subgroup of G, AG = <1>, and M a minimal normal subgroup
of G. Then
i) M = CG(M);

ii) M is the only minimal normal subgroup of G;
iii) G = A M and A n M = <1>;

iv) if Go/M is a minimal normal subgroup of G/M and A = A n Go,

0
then A =NG(A0), ]G0] = ]MHAO], ]M] = pa, ]A0] = qB, and
p # q for p and q primes.

Theorem L ]2]: (W. Gasscthz) Let 1 = Nr S...S N0 = H be a
composition series of the solvable group H. If M is a group of
automorphisms such that each subgroup of the above series is left
fixed by M then M is solvable.

Theorem M ]10]: The automorphism group of a cyclic p—group, p

a prime, is cyclic of order ¢(pn) = pn-1(p-l), pn is the order of

 

 

 

52

the group and ¢ is the Euler ¢-function.

Theorem N ]3]: (Burnside) Every group of order pan, p
and q .primes, is solvable.

Theorem 0 ]3]: Let ¢ be a fixed-point-free automorphism
of G. Then every element of G can be expressed in the form x-lx¢
for Suitable x in G.

Theorem P ]3]: (Schur-Zassenhaus-Feit-Thompson) Let H be
a normal Sn-subgroup of G. Then we have
i) G possesses an Sn,-subgroup K which is a complement to H

in G.
ii) Any two Sn,-Subgroups of G are conjugate in G.

Theorenng ] l]: Gdielandt) Let H* be a normal subgroup of
H which is a subgroup of G Such that H n Ht S H* for all
t E G - H. Then there is a unique solution G* of the equations
G* A G, H n G* = H , G = H 6*. Indeed G* coincides with the sub-

*
set of G given by S = G - U (H - H )8.
SEC
Theorem R ]3]: (Burnside) If a Sylow p-subgroup of G lies

in the center of its normalizer in G, then G has a normal p-
complement.

Theorem S ]3]: Let H be an Abelian Hall subgroup of G with
H S Z(NG(H)). Then G possesses a normal H-complement.

TheoremT ]l]: (Wielandt) Let H* be a normal subgroup of
H which is a subgroup of the group G, and set F = NG(H). Assume
that ([F : H], [H : H*]) = 1 and that H n Ht S H* for all
t 6 G - F. Then there is at most one solution G* of G* A G,
H n G* = H*, H 6* = G. Such a solution exists if and only if there

* * * * *
isasolution F of H SF AF and [FzF]=[H:H]. This

 

 

 

53

latter set of equations has at most one solution.

Theorem U L6]: (P. Hall) For i = 1,2,...,n, let
prime divisors of the order of a finite solvable group G;
has n subgroups Pi so that Pi is a Sylow pi-subgroup

and so that, for i, j, = 1,2,...,n, Pin = Pj Pi.

pi be the
then G
of G

 

 

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