groups whose homomorphic images have a transitive normality relation

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 176, February 1973

GROUPS WHOSEHOMOMORPHICIMAGES ■ HAVE A TRANSITIVE NORMALITYRELATION BY

DEREK J. S. ROBINSON(l) ABSTRACT. A group G is a T-group if H ...>

hence

is metabelian.

of G; the latter

so

p such

P O G, and since

is an infinite

and

is impossible

of nontrivial

contain

normal

Pa> ...,

since

L— is abelian.

that the subgroup

P cannot

Thus

P = \a; a e

a minimal

subgroups

normal

of G,

(a LD/D ^ L, so G/z9 is a soluble

T-group

in view of the structure are isomorphic isomotphic

Then

L/M

in L and thus

must

agree

with

must

1, which group fully

in L/M;

a —►a~

shows

C/M

a —> a~

automorphisms

induce

that

of C is normal on L. Since

T-groups

g induces Since

elements

in C/M.

C is abelian

the group

a powet L/M

has

of the same

The intetsection

and

in G. Lemma

where

ag = a~

G is metabelian,

L and

Let

a in L LD/D

L to be

M be a subgroup

Z is the subgroup g oí G\C automotphism

in C/M

of evety

order

to the same

of all subgroups

[zL, X ] = 1; therefore

a —► which

finite

order

power,

like

a e C. Hence

a contradiction,

of all in-

induces

elements

for all

6 now gives

for all

now forces

numbets.

An element

also

map

also

Q/Z,

is abelian.

on L/M.

ae = a~

irreducibility

of rational

with

then

of type I and because

Rational

group

is isomorphic

M < G and

I. If g e G\C,

groups.

Q, the additive

a~

and power

of soluble

as G-operator

with

of L such that tegers.

of type

g

M is

every

so X acts

subfaith-

X is abelian

and G = L. Let

1. Since

N ~ A/C/C,

is impossible,

abelian

of groups

N, but not

A/C/C—and

hence

Horn (G'/N,N) G

to a = 0

is abelian.

R =^ G'/N. and

Clearly

R n Dj = (t, G') r\ Dx= (t2, G') n Dy

/ by tó, we can assume

unless

of G/C

are congruent

so W O Dj = (a).

G' or f2 / G'. Also, a ¿in

= G/C.

Dj). Also

lfnDr

ing

of W. First

= a2« 8

and

zze = a2

2

.- a

g e G; 8a.

/ 1 mod 4.

£ G', then in addition

D / 1; for

thus W is not a JNT-group by Lemma 6 and D / 1.

no element

G is of type VI(a) or (b) according

of order

as í

p except

£ G or t

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

«j

since

4 G'.

/V is the mono-

1973]

HOMOMORPHICIMAGES AND TRANSITIVE NORMALITY

The remaining exactly

possibility

2. The centre

under

of C is

heading

(5.332)

C/N

is abelian;

Z and

is that

201

C is nilpotent

of class

thus

N = CA G'< Z < C. If x, y e C, then mentary

abelian

1 = [x, y]^ = [xp, y], showing that Cp < Z and C/Z

Let us prove

next that

abelian,

we can write

x = cA

for some

G; then

cs = c

Cp = G'.

Cp = G x F.

c £ C and

Now

G' < Cp is immediate,

Suppose

x is a nonttivial

a £ N; for C/N

8¿) for some

is abelian.

(cp)

< G. Now

fore

x = cA

a / x since

e (cp)

and

and since element

Let

Cp is

of F; then

g be any element

of

b £ N. Hence

(c")g Ar"8^ and

is an ele-

p-group.

Ar^8

G' O F = 1; thus

(x) O G. This

gives

cp / 1 and

a e (A).

the contradiction

There-

N < (x).

Hence

F = 1 and Cp = G'. Let

Sx Z: À e Aj be a basis

fot the elementary

X = (xx: X e A). Then C = XZ = XG'Dy y ~ x\

x\

ments

a wnere

of A. Then

x^

a e C,

Writing F = XG we obtain xp = ap.

that

This

n. are integers

• • • x^r £ Z and the lineat

plies that p\n. for i = 1, •••, Suppose

the

p-group

C/Z

and the

À. ate distinct

ele-

independence

of the

Z im-

for all

z and

that

G" = Cp, there

(x^a~

y = x»

r

is an element

= 1. Replacing

x.

p-group.

Since

and

in which

the centre

G

•• •x,7' mod N whete r

Next

the

N is actually

a oí G

by x^a~

X d G = N = (a ), the group of X and

now the position

(a.)

of / ; if /

and

[x^ , t] / 1 for some

x^ = *xa,;

some

thus

x e X (note

(/xx )

À € A. Since

= a.x^=

N = X ); then

(¿x)

x^N

that

we may assume

XHZ;

since

Thus

z

of X since

p\n . z

£(X)

Consequently

X is an extra-

E is a direct

product

of X

are amalgamated. 4 G , then

p = 2 and

u £ D v li t2 £ G' and /2 / 1, then p = 2 and /2 = a then

such

À. are distinct. the centre

X' = £(X) = N and X/N ~ C/Z.

special

Consider

1

y e N as required.

< X n Z = N. Thus

x,

C= E x D .

that xp = 1 for all À e A. This implies that X n Z = N: for let ye X' < N, we can write

and set

and write

r. Hence y £ CPG = G' and y e G A D = 1.

xp / 1. Since

implies

abelian

Let y e (XG') nD,

has order

1. If, however,

z: = a u where

Suppose that

2, it is centralised

[X, z] = 1, write

= 1. Thetefore,

[X, t] / 1;

if /

by t

a^ = x

for

e G*, we may assume

that t2 = 1. From C = XZ and equations

define

(12) and (17) we obtain

G = (/, X, y, Dj).

Now

W= (/, X, Y). Writing

Dj = «i, G') h DAxD, we have G = WD and W< G. Suppose that w £ W CI D and, using


G* < Y, write

202

D. J. S. ROBINSON

w = Axy where that

x e X and

i may be assumed

y £ Y; then

even.

w £ (t, G') n D = 1. Hence

Since

y £ T n

t

Y =G'

and

£ Z, it follows

that

of W. First

If t2 £ G', then W n Dj = 1; otherwise F or F x (a)

according

is isomorphic

to

1 modulo

or

au

Finally

a

g

then

faithfully

soluble

nilpotent

J NT-group

subgroup

of G contained

for some prime

of p-adic

integers

all of which

on W and centralises to whether

is both

a

and

are congruent

C abelian)

^ 1 mod 4. Since

s

U-O C =

W/W C\ C ~ G/C

that

zzß = u

= 1 mod p for all

g

^

in G and [X, G] < N.

/NT-groups.

which

Moreover

(as in the case

= 1 mod 4 or a

or (b) according

(t2, G') n Dy

t2 = a2« and W H Z> = 1; then

in P/N

N < Pp and consequently

b8 = bac that

exponent

for some

a8 = aa

2

; therefore

pe, the congruence

an automorphism

c £ N, and a

7

=1

a = 1 mod p.

mod pe implies

of p; therefore


kP

a8 = ib8)p = bap = aa.

= a mod p and

ap

of order a power

a = bp for some

that

g £ P by (23).

g

1973]

HOMOMORPHICIMAGES AND TRANSITIVE NORMALITY

If, however,

with

P/N

has infinite

exponent,

a = 1 mod p and p > 2, implies

a must have finite

that

(25)

a = 1. These

order;

205 this,

arguments

together

indicate

that

Pp = 1. Next the centre

ax £ Z; then C(Aal

(a

)~>P\

of P will be identified;

N) < G, so a^ < (a^,

therefore

Maschke's

call this Z. Clearly

N), which implies

theorem

can be applied

that

N < Z. Let 1 / a

to a

is finite.

; in the

Now

usual

way

it follows that aGx= N. Thus Z = N. Now P/N

Choose

is elementary

a basis

for P/N,

abelian

say

by (25); hence

p-group.

ixxN: À e A}. Then

(26)

[xx, x^] = a/(x,ii)

where

/ is a nondegenerate

order

q dividing

order

q such that

form

x —> x"

alternating

p — 1 since

g induces

G = P(g)

and

in the elementary where

bilinear

P = CGÍP/N).

group

simplify

integers these

rex satisfying

equations.

is a p. £ A such

that

ment of the form x.

xxa"

where

be solved

that

A brief

computation

(g)

acts

with

g with

faithfully

on P.

automorphism

change

of the

of basis

will

/ is nondegenerate,

replace using

thete

xx by a suitable

ele-

(26) and (27) yields

p). We wish to show that

s ß 0 mod p and

(28)

is cyclic

UeA), n^ / 0. Since

+ sti2)-fi\,

t with

G/P

is an element

a power

0 < 7ZX< p. A suitable

Suppose

u = sn^ + tn

for s and

P/N

/(À, p) /= 0 mod p. We shall = x^x'.

Now

there

Thus

xl = xy\

for certain

form.

Hence

P n (g) = 1. Moreover abelian

1 < « < p.

(27)

t é 0 mod p. This

x8 =

a = 0 mod p can

amounts

to solving

X72 + y?2 + z = 0 mod p

fot x ¿ 0 mod p and

Since

y £ 0 mod p where

72x ^ 0 mod p, we need

72 = 0, any

Now replace this

operation

z = (")/(A,

p > 2. Consequently

(28) has

xx by x"x , observing

whenever

necessary,

Also

usual,

T and is not abelian,

(by Lemma

2.4.1

L = Ve', G], Define

< 2. Moteover

G/C

has

then

of [18])

L/N

a basis

1 or 2 because

that

z £ 0 mod p.

+ z é 0 mod p. If 1 < y < p and of the required

for P/N.

for which

yn sort.

Performing

x^ = xx for all

If a soluble p-group has the

p = 2 [18, Lemma

C = C AG'/N)

order

a solution

at a basis

is a radicable

yn

y so that

that we retain

we arrive

A e A. Thus G is of type V. (6.22) Case [N, G'] = 1 and P/N nonabelian. ptoperty

zj); notice

only look for a y such that

y ^ 0 will do; if ra /= 0, we can choose

+ 22; 0 mod p since

finite

P is an extra-special

4.2.l]. abelian

and note that

C/N

+ 1 are the only

order.


Thus 2-group

P is a 2-group. where,

is nilpotent 2-adic

as

of class

integers

with

206

D. J. S. ROBINSON Let

x £ C n P; then

[February

aN —» [a, x] is a homomorphism

[A/, C n p] = 1 (see equation

(19)). But HomÍL/'N,

(29)

of L/A/ into

N since

N) = 0; therefore

[L,CnP]=l.

Since L < G < C O P, it follows that L is abelian. Our next ments

aim is to prove

of G with

elements

odd order

in C/N

which

Assume that Q/N. group

that

G is a 2-group.

belong

have

to C. Now

odd order

Since

C/N

forman

|G: C| = 1 or 2, all ele-

is a Dedekind

abelian

group,

subgroup,

so the

say

Q/N.

From this it follows that [N, Q] / 1; for if [A/, Q] = 1, the

Q is nilpotent,

and,

since

0 N; then L/'N is a nontrivial that

radicable

abelian

L is radicable.

2-group and L

Now commutation

p 1; therefore

L. L~ is radicable,

with a fixed element

of C induces

of L into A/, and yet Hom(L, N) = 0; thus [L, C] = 1 and in particular

contradiction.

It follows that L = N and G/A/ is Dedekind;

and (29) becomes

[A/, Q] = 1, a

thus

C = GAG' N) = G o

[L, P] = 1. Now [A/, Q] =■1 implies that G splits

Lemma 8; say G = NX and N C\ X = 1. Therefore

which shows

a homomorphism

over iV, by

P = P n ÍNX) = NÍP n X). Since

P//V is not abelian, P n X /= 1. Also [/V, P n X] < [L, P] = 1, so P n X O A/X= G. Thus

G/P

that

n X is a T-group

\N\ - 2. Consequently

tradiction

establishes

isomorphism

N < £(G)

that

which

Observe

that

G/N

(30) would

L > N and this, G/N

Lemma

each

a

element

that

of C/N

means

and

with

the

that

Equation

C is nilpotent

on L/N

group;

seen, property

and

t

at most

a - 1 e Hom(L,

(31) which

shows

that

z centralises

that

also

G = (C, /)

£ C; also, with

2. Let

nonnilpotent.

where

of course, the

C = G,

L is radicable.

C/N

is abelian of C/N,

o be the automorphism

N) = 0. Therefore

r = a and

By

t transforms

commutativity

by t. Then

N. Since

con-

L is radicable:

of L induced

«' = a-1,

This

L = N and

Also

T and it is

together

r for the automorphism

r"

implies

one can write

(30),

of class

for if it were,

G is nilpotent.

been

into its inverse

exponent.

of L and write

trivial

2-group

of [18] this

and of infinite

already

G is nilpotent.

(29) now yields

C = CGÍL).

[L, G] = 1, i.e.

as has

is a soluble 4.2.1

implies

is not a Dedekind

become

that

Equation

[L, C] = 1 and

so that

N ~ A/(P n X), P D X shows

implies

G is a 2-group.

(30)

for

and the

a —►

r ~ a is

(a e L), G = (C, /). equation

(30) permits

us to conclude

that [N, Gj = 1. Suppose there exists Al < G with M ■/ 1 and M O N = 1. Since AI~ A1A/A',

one can

assume

and its structure abelian.

Now

that

Al has order

is similar / transforms

to that elements

2. Also of G/N.

L < Al, so G/A1 is not a Dedekind In particular

of L into their

CM/M—and

inverses.


hence

It follows

group C-is

that

a

=

1973] a~

HOMOMORPHICIMAGES AND TRANSITIVE NORMALITY for all

monolith

a £ C, which

equal

Define wtite

is impossible

to N. This

Z = L x D. Suppose

is in the centre

Thus

6. Hence

L is of type

2

L < Z and,

D contains

G is monolithic and

an element

1 /

abelian.

(d ) < G; this

This

implies

DN/N

2.

tadicable,

d of order

we may

4; then

dl = d~

d2 £ Z, it follows

is impossible

that

with

N has order

L being

id2)1 = id~ a)2 = d~2 = d2. Since

of G and

D is elementaty

that

of C. Then

that

for some a in N. Hence d

indicates

Z to be the centte

by Lemma

207

since

lies

id2)

a

that

H N = 1.

in the centre

of G/N,

so that

(32)

[D,G] [x, d] is a homomotphism

= 1 = [N, G]. Now CD(G)< 2, we must have Consider

ators

a)-

a = c.

the position

L.

of / . Let

If c £ C, then

On account

(t2) < G. Therefore, centralised

either

for t

t

be normal

(6.221) Case C abelian. dl = a^d, we have Thetefote

c - c~

has order

that

Also

dividing

or ad

[C, D]

t

set of gener-

a in N. Hence £ £(G);

t2 £ Z and since

4. Thus

/

cl

=

in particulat

t2N is

e (a ) x (z/) and

(if d/ 1); for if d / 1, then dl = a d by

in G.

Here C = Z - L x D, and aA 1 by Lemma 6. Since

itd)2 = t2a d2 = t a..

we can assume

a for some

it follows

z"2= 1 or N < (t2).

are 1, a

(a1) cannot

a , a , ■■ • be a canonical

of G = (C, r)

by /, the element

the possibilities (32) since

into N since

Horn ÍG/C, N) has order

\D\ = 1 or 2. Write D = (d).

next

for the 2°°-group

(c~

of G/C

G, so CD(G) = 1, and, since

that either

/

Hence = 1 or /

t

= a

implies

= a d; the

that

order

(fa?)2 = 1.

of t is

2 or 8.

Thus G is of type II.

(6.222) Case and

y belong

abelian.

and

C nilpotent

to C, then

Choose

a basis

z = 0 or 1. Now

x~A= xxfc_ ; then follows

that

can