Abelian Groups - László Fuchs [Springer]

2 Automorphism Groups of p-Groups 663
(d) If an involution of A is extremal, then for every commuting involution ,
in the direct decomposition stated in Sect. 1 (h) there are at most 3 non-zero
summands.
Example 2.2 (Leptin [2]). It can happen that, for different primes p; q, ap-group and a q-group
have isomorphic automorphism groups, though this is a very rare phenomenon. This is the case if
and only if the groups are cyclic: Z.p k / and Z.q/ such that 3 p < q and q 1 D p k 1 .p 1/:
(E.g., q is a Fermat prime, and p D 2.)
The main result on p-groups is the following theorem which we state without
proof. It is an important contribution to the theory of p-groups.
Theorem 2.3 (Leptin [2], Liebert [5]). Assume p >2and A; C are p-groups. If
Aut A Š Aut C, then A Š C.
ut
F Notes. Automorphism groups of finite abelian groups have been investigated by several
authors, starting with Shoda [1], and as a result, a lot is known about them. A further step was taken
by Baer [3] who studied the normal structure of Aut A for infinite p-groups A; he also investigated
the correspondence between characteristic subgroups of A and normal subgroups of Aut A. That
the results gave special status to the prime number 2 should come as no surprise. Later, the study
of the normal structure of Aut A has advanced considerably, see Faltings [1], Freedman [1], Leptin
[3], Mader [1, 2], Hill [10], and above all Hausen [1, 2, 3].
The amount of open questions as to how the normal structure can be described is staggering.
Several papers were devoted to the question of characteristic subgroups whose automorphisms
extend to automorphisms of the containing group, as well as to pairs of subgroups whose
isomorphism extends to an automorphism of the entire group. Solvable automorphism groups were
studied by Shlyafer [1] and Brandl [1]: for a p-group A,AutA is solvable only if A has the minimum
condition on subgroups. See also papers by Hausen listed in the Bibliography.
The multiplicative analogue of the Baer–Kaplansky theorem is more subtle; it has not been
completely settled: the case p D 2 is still open. The work started with Freedman [1] who proved
that for p 5 two countable reduced p-groups A; C are isomorphic whenever p ! A Š p ! C and
Aut A Š Aut C. It was a big step forward when Leptin [2] succeeded in relaxing the requirements
of countability and the isomorphy of the first Ulm subgroups. In cases p D 2; 3, involutions
created serious problems. Only 25 years later was able Liebert [5] to find a proof, using completely
different methods, that included also the prime p D 3. Unfortunately, the proofs do not provide a
method to encode relevant structural information about the group from the automorphism group.
Schultz [3] claimed a proof for p D 2, but his proof relied on a lemma that held only for p 3.
The case p D 2 seems awfully difficult, and the Leptin-Liebert theorem might not extend to this
special case. For p 3, Liebert [5] describes the possible isomorphisms between Aut A and Aut A 0
for isomorphic p-groups A; A 0 ; there exist some not induced by any group isomorphism.
Leptin [2] has an interesting in-depth study of commuting involutions. For an illuminating
survey on results concerning automorphism groups up to the year 1999, we refer to Schultz [4]
(the claim for p D 2 should be ignored).
The elementary equivalence of automorphism groups of p-groups .p 3/ were discussed by
Bunina–Roǐzner [1].
Exercises
(1) (a) An elementary group of order p m has .p m 1/.p m p/ .p m p m 1 /
automorphisms.
(b) j Aut Aj D2 if A is an elementary p-group of cardinality .