DEREK J. S. ROBINSON(l). ABSTRACT. A group G is ... D. J. S. ROBINSON. [February. [X,.+ 1Y] = [[X; . ...... Chelsea, New York, 1960. MR 15, 501; MR 22 #727.
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
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«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
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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
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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.
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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.
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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