PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 48, Number 1, March 1975
AUTOMORPHISMGROUPS OF ABELIAN p-GROUPS
JUTTA HAUSEN1
ABSTRACT.
duced
abelian
Let
subgroup
of
group
T of rank
of
1. The
abelian
V does
every
mal subgroup
of r
the following,
then
not contain
possess
(i.e.
contains
noncentral
abelian
re-
normal
normal
G is a reduced
p-sub-
p-primary
of all automorphisms
the normal
normal
structure
p-subgroups
nontrivial
noncentral
of a nonelementary
every
elementary
V is the group
abelian
V does
group
that
2.
Throughout
If G is elementary
pG 4 0 then
a noncentral
at least
p > 5, and
In particular,
this case,
p > 5. It is shown
T contains
result.
group,
T be the automorphism
p-group,
normal
of V is well
Moreover,
in the center
normal
known.
4 1 [2, pp. 41, 45].
p-subgroups.
not contained
a noncentral
of G.
If
in
ZY of F) nor-
p-subgroup
N of Y such
that Np = 1 [6, Theorem A].
The purpose
siderably
Theorem.
abelian
of this
Let
p-group,
a noncentral
is to prove
p > 5. Then
normal
group
a [noncentral]
socle
is the center
of D ,). Whether
Notation
otherwise.
in the endomorphism
Note that mappings
cyclic
is con-
the dihedral
2-group
normal
subgroup
reduced
of Y contains
of Y of rank at least
since
the above
Received
and terminology
Calculations
(namely
subgroup
Theorem
ring of G. The
are written
by the editors
will be that
involving
group
holds
D4 oc-
G = Z(2)
of order
2.
©
4 whose
true
for p = 3
following
of [3] and [6] unless
automorphisms
facts
are used
are carried
out
frequently.
to the right.
December
AMS {MOS) subject classifications
Grant
which
question.
2. Tools.
groups
normal
of an abelian
D , contains
Secondary
result
of a nonelementary
p-subgroup
p 4 2 is indispensable
Z(4);
is an open
group
every noncentral
abelian
as an automorphism
explained
the following
Y be the automorphism
elementary
The hypothesis
curs
note
stronger.
26, 197 3.
(1970), Primary 20K30, 20K10, 20F15;
20F30.
Key words and phrases.
Abelian
p-group,
automorphism
of automorphism
groups.
'This
research
was supported
in part by the National
(jF-34195.
group,
Science
normal
Foundation
subunder
Copyright © 1975, AmericanMathematicalSociety
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67
68
JUTTA HAUSEN
(2.1)
The center
plications
with units
G is unbounded
of order
of Y. The center
in the ring
R
Zr
and the center
then
cü R
precisely
integers
[1, pp.
mapping
characteristic
111].
If
if and only if So. = S for all
Let fix (S/A)
if and only if G is
in ß/A
where
in G then
be the set of all y eT
A < B are subgroups
fix (ß/A)
is a noimal
A ^
Horn (G/A,
(2.3)
The normal
such that
A). In particular,
subgroups
Np = 1. Then
stabilizers
of exponent
N < 1* where
inducing the
of G. Ii A and
subgroup
of A in G is defined as stab A = fix A O fix (G/A).
of r
110,
no elements
cyclic [3, p. 222].
(2.2) Stabilizers.
stab
of the multi-
of Y contains
S oí G [l, pp. 110, 111]. F is commutative
(locally)
identity
oí p-adic
P
a of G is central
p. An automorphism
subgroups
of Y consists
of Y. The
B are
stabilizer
It is well known that
are abelian.
p. Let
N be a normal
W consists
of all
subgroup
y £ Y such
that G[p](y - 1) < pG and p(y - 1) = 0 [4, pp. 409, 410]. If ifj £ W then
G(ip- 1) <G[p] and (i/j ~ l)3 = 0 [4, p. 411]. Hence f < fix (pG) n
fix (G/G[p])
and, since
An immediate
Lemma
2.4.
N n fix G[p] and
p > 3, *" = 1.
consequence
is the following
// N is a normal
N n
fix (G/pG)
p-subgroup
result.
of Y such
are elementary
that
abelian
Np = 1 then
normal
p-subgroups
ofY.
Proof. By (2.3) N < V, and «Pn fix G[p] < stab GÍp], V n fix (G/pG) <
stab pG. By (2.2)
stab G[p] ~ Horn (G/G[p], G[p]),
which
are elementary
The following
Lemma
0^
2.5.
pmG for some
stab pG » Horn (G/pG, pG),
abelian.
two lemmas
Let
are technical.
G = A © (Â) where
integer
that Np = 1. If there exists
m>l.
Let
A has
rank at least
N be a normal
subgroup
two and
pmA =
of Y such
y £ N such that (y - I)2 4 0 then N n fix G[p]
is noncentral.
Proof. Let y - 1 + r £ N such that r2 4 0. Then Gr < G[p], pr = 0 (cf.
(2.3)),
and
of maximal
r
4 0 implies
order,
Gr ¿ pG.
one can assume
G = (a) © B © (h),
Suppose
hr
Since
G is generated
4 0. Hence
hr=a,
ar 4 0,
pß = 0. Then pG = (pA) and (pmh) >(pG)[p]
Hence ar 4 0 and B 4 0 imply
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by its elements
hr £ G[p]\pG
and
ß ^ 0.
> G[p]r > ((a)©B>.
AUTOMORPHISM GROUPS OF ABELIAN p-GROUPS
[a; © B = (a) © K,
69
Kr = 0 4 K,
In this case, pick any 0 4 x £ K. If pB 4 0 pick 0 4 x £ (pB)[p] and
put K = B. In either
case
G=(a)®K®(h),
Define
the
endomorphism
04x£K[p],
xr=0.
a of G by
ao = x,
Ko = 0,
ho = 0.
Then a2 = 0 and Gor = (x)r = 0. Lemma 2.6 of [7] implies
8 = 1 + to £ N.
From h(8 - 1) = hro = ao = x 4 (h) it follows that 8 4 ZY (cf. (2.1)). Since
G[p](8-
1)= G[p]ra < pGcr = p • (x) = 0,
8 £ N C\ fix G[p], completing
Lemma
2.6.
Let
the proof.
G and
N be as in Lemma
2.5 and suppose
that (y-1)
= 0 for all y £ N. If N n fix Glp] < ZY and N n fix (G/pG) < Zr then N < ZY.
Proof.
N\ZY.
sequently
(2.7)
Assume
Then
by way of contradiction
y 4 fix (G/pG)
hr £ G\.p]\pG
and,
there
exists
one can assume
y = 1 + r £
hr 4 pG,
con-
and
G = \a) ® B © (h),
By hypothesis
that
as above,
hr = a,
0(a) = p,
y ¿ fix G[p] and hence
© (p¿>) and or = 0, pr = 0, this
ar = ¿>r2= 0.
G[p]r=^ 0. Since
implies
the existence
G[p] < (a) © Bip]
oí b £ B[p]\pB
such
that
(2.8)
Let
W
k = 2~ (p + 1). Since
p is odd,
tively prime. Using (2.7), define
¿>r/3 = br since
k is an integer
and
k and
p are rela-
ß £ Y by
aß = ka+b,
Note that
0.
br £ pG <B
(B®(h))(ß-1)=0.
@ (h).
One verifies
2(fl - b), and hence
«yS-1r/S = 2(a - b)rß = -2£Vß = -2br,
bß~lrß
= èr/8 = br,
hß~lrß
m hrß *=aß = ka + b.
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that
a/3-1
=
70
JUTTA HAUSEN
Let 8 = yß~lyß
= (1 + r)(l + ß'lrß).
V = r+
rß~lrß.
ß~lrß+
Then ¿5 £ N and 5 = 1 + r¡, where
Since
h(8 - 1) = hr¡ = hr + hß-Xrß + hrß'lrß
» a + (ka + b) + ,(-2br) = ,(k + l)a
and br £pG,
it follows that h(8 — l) 4 (h).
(2.1)) and, by hypothesis,
pGß
+ b - 2br
Hence 8 £ N is noncentral
(8 - l)2 = 0. From a, b £ G[p], G[p]rß-lrß
(cf.
<
rß = pGrß = 0, and r2 = 0, one obtains
0 = h(8 - l)2 = [(k + l)a + b - 2br]r¡
= [(k + l)a
+ b - 2br](r
= [(k + l)a
+ b](r
= (k + l)(-2br)
+ ß~lrß
+ ß-hß
+ rß~lrß)
- {k + l)a(r
+ ß~lrß)
+ b(r +ß~1rß
+ br + br
= -2kbr = -2 • 2-1( p + l) br = - br.
Hence
br - 0, contradicting
Corollary
then
2.9.
G and
3. Proof.
Assume
subgroup
the situation
of Y. It was
central.
Hence,
it suffices
tary abelian
normal
p-subgroup
normal
nent
subgroup
p. This
// N is noncentral
is noncentral.
and let
in [6] that cyclic
to show
that
a noncentral
N be a noncentral
normal
N contains
of Y. By Theorem
subgroups
a noncentral
A of [6], every
normal
subgroup
of Y
elemen-
noncentral
of Y of expo-
the assumption
(3.1)
Np = 1.
Distinguish
the following
Case
cases.
1. G is unbounded.
(2.1)) and every p-subgroup
and Lemma
2.4, it suffices
Then
contains
no elements
4 1 oí Y is noncentral.
Therefore,
to prove
ger, 2re = stab G[p"]. Then £
showing
Nnl
4 1 for some
N n S^ = 1 for all
this
2.5.
of the Theorem
shown
of' Y contains
permits
the lemma.
N be as in Lemma
N n fix G\_p] or N n fix (G/pG)
normal
ate
Let
(2.8) and proving
implies
n > 1. Since
N is contained
Zr
N n fix G[p] 4 1. Let,
of order
using
p (cf.
(3.1)
for n > 1 an inte-
< fix Glp] and the proof will be completed
n. Assume,
N
and the
in the centralizer
by way of contradiction,
S
are normal
CS
n
of 2
n
n > 1. By Lemma 2.1 of [5], Cln < ZY ■ fix GÍpn]. Hence
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subgroups
that
of Y
in Y, for all
by
AUTOMORPHISM GROUPS OF ABELIAN p-GROUPS
(3.2)
N < H Cln < fl (ZY ■fix Glp"]) = <D.
n>\
Using
71
(2.1)
induces
n>\
and the fact
the identity
that
G = \J
mapping
>1 G[p"],
in the lattice
one verifies
that
of all subgroups
(f> £ ZY. This together with (3.2) implies
every
(p £ $
of G and hence,
N < ZY which is the desired con-
tradiction.
Case
group
2. G is bounded.
with
tersection
Dr
It has
two independent
DT
of all noncentral
is an elementary
one can assume
p-group
G = A © (h)
in [7] that
order
subgroups
G is a bounded
if and only
if the in-
of Y is noncentral;
if pG 4 0 [7, Theorems
and
Ii A has rank at least 2, Corollary
Suppose
proved
of maximal
normal
abelian
that
been
elements
pmG 4 0 = pmA
2,3].
for some
and
Therefore,
integer
m >1.
2.9, Lemma 2.4, and (3.1) finish the proof.
that
(3.3)
G=(a)®(h),
0(a) < 0(h) = pm+l.
If pa 4 0 then G[p] < pG and (3.1), (2.3), and (2.2) imply
N < »P < stab G[p] =* Horn (G/G[p], G[p]).
Hence
N is elementary
It remains
and (2.3),
abelian
to consider
and the proof
the case
N < 1J and the elements
where
is completed.
a in (3-3)
has
order
in W can be identified
p. By (3.1)
with matrices
of
the form
"l
pmk
J
where
1 + Pmn.
0 < k, I, n < p - I are integers.
N - *P or N has order
tary abelian
order
p
abelian.
groups
normal
of r
*P has
p or p . If N = *P then
p-subgroups
or p, then
(Actually,
Hence
are contained
N contains
of Y, for instance
N is commutative
the case
and,
stab
because
\N\ = p cannot
in Zr
order
occur
[6, Theorem
and either
noncentral
elemen-
G[p] < *P. If N has
of (3.1),
since
B].)
p
N is elementary
cyclic
normal
The Theorem
sub-
is proven.
REFERENCES
1. R. Baer,
Primary
abelian
groups
and their
automorphisms,
Amer.
J. Math.
59
(1937), 99-117.
2. J. Dieudonne,
Les
France 71 ( 1943), 27-45.
3. L. Fuchs,
Internat.
Ser.
Abelian
Monographs
determinants
sur un corps
non commutatif,
Bull.
Soc.
Math.
MR 7, 3groups,
on Pure
Akad. Kiado, Budapest,
and Appl.
Math.,
Pergamon
MR
21 #5672; 22 #2644.
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
1958; republished
Press,
New York,
by
1960.
72
JUTTA HAUSEN
4. J. Hausen,
On the normal
structure
of automorphism
groups
of abelian
p-groups, J. London Math. Soc. (2) 5 (1972), 409-413.
5. -,
The automorphism
normal subgroups,
6. -,.
groups,
Trans.
group
of an abelian
p-group
and its noncentral
p-group
and its normal
J. Algebra 30 (1974) , 459—472.
The automorphism
group
of an abelian
p-sub-
Amer. Math. Soc. 182 (1973), 159—164.
7. -,
groups of abelian
Structural
relations
p-groups,
Proc. London Math. Soc. (3) 28 (1974), 614—630.
between
general
linear
groups
and automorphism
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF HOUSTON, HOUSTON, TEXAS
77004
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