Abelian Groups - László Fuchs [Springer]

662 17 Automorphism Groups
Accordingly, from Theorem 8.6 in Chapter 18 it follows that for the center
z.Aut A/ of the automorphism group of a p-group A we have: if p >2;then
z.Aut A/ Š Z.p 1/ ˚ J p or Š Z.p 1/ ˚ Z.p m 1 /
according as A is unbounded or bounded by p m , while if p D 2,then
z.Aut A/ Š J 2 or Š Z.2 m 1 / or Š Z.2/ ˚ J 2
according as A is unbounded, bounded by 2 m , or of exceptional type.
Since a group is abelian exactly if it coincides with its center, from the preceding
theorem it is immediate that the automorphism group of a p-group is commutative
if and only if it is cocyclic or isomorphic to Z.2 1 / ˚ Z.2/:
Involutions of p-Groups It appears that involutions enter the picture naturally.
Involutions contain a lot of information about p-groups, but we have to learn how to
read them. We now go on to collect some relevant observations about involutions in
p-groups.
If S Aut A,
c.S/ Df˛ 2 Aut A j ˛ D ˛ 8 2 Sg
will denote the centralizer of S in Aut A. Of course, c.Aut A/ D z.Aut A/. An
involution is said to be extremal if either A C
or A
is indecomposable ¤ 0.
In (a)–(e) we keep assuming that p ¤ 2.
(a) The p-group A is bounded, and p n is the l.u.b. for the orders of its elements if
and only if z.Aut A/ Š Z.p n 1 .p 1//.
(b) If A is a p-group, and if 2 End A is such that Ker D 0 and .p n AŒp/ D
p n AŒp for all n 1/. By induction hypothesis, there are a b 2 AŒp n such that
.p n 1 b/ D p n 1 a,andac 2 A with .c/ D a .b/. Hence a D .b C c/,and
is epic as well.
(c) Assume A D C 1 ˚˚C k .C i ¤ 0/ is a direct decomposition of the p-group
A. Let i denote the involutions that belong to this decomposition. Then the
centralizer
cf 1 ;:::; k gDAut C 1 Aut C k ;
and if k 3, then the center of this centralizer contains exactly 2 k involutions.
That under the identification stated in Sect. 1 (d), all Aut C i belong to the
centralizer of the set of the i is evident. Conversely, if ˛ 2 Aut A commutes
with each of i ,then˛C i D ˛A i
A i
D C i for every i, whence ˛C i D C i ,so
˛ 2Aut C i : The final claim is evident.