Throughout the following G
denotes an abelian pgroup (for some fixed prime p) and A(G) its automorphism
group.
The subject of this article is the question to what extent the structure of G and
the structure of A(G) determine each other. Theorems of the type: G is a Pgroup if
and only if A(G) is a Qgroup, where P and Q are group theoretical properties, have
been proved by R. Baer; some others are wellknown.
This paper gives a characterisation of this kind for the class P of all abelian
pgroups whose homogeneous direct summands have finite rank (an abelian pgroup
is called homogeneous if it is a direct sum of isomorphic subgroups of rank
1).
For this purpose it is convenient to define, for every group X, a characteristic
subgroup ΩX, called the hypo residuum of X. This is the product of all
normal subgroups N of X, such that every finite epimorphic image of N is
trivial.
The main result is the following
THEOREM A. Every homogeneous direct summand of G has finite rank if and
only if ΩA(G) = 1.
A consequence of this theorem is the following fact: if A(G) contains a quasicyclic
subgroup, then the group of all permutations on a countably infinite set (and hence
every countable group) is isomorphic to a group of automorphisms of G. This
happens (if and) only if G possesses a homogeneous direct summand of infinite
rank.
The group Γ(G) of all automorphisms γ of G inducing the lautomorphism in
Glp^{ω}G is of special significance. It is shown that ΩΓ(G) = 1 for every reduced abelian
pgroup G (Theorem 2) and furthermore
THEOREM B. If G is reduced then ΩA(G) = 1 if and only if A(G)∕Γ(G) is
residually finite.
Closely connected with the concept of the hypo residuum of a group X is the
descending chain of the so called higher residua ℛ_{μ}X of X: let ℛ_{0}X = X, define
ℛ_{λ}X = ⋂
_{μ<λ}ℛ_{μ}X if λ is a limit ordinal and ℛ_{μ}X has been defined already for all
ordinals μ < λ, and let ℛ_{μ+1}X be the intersection of all subgroups of ℛ_{μ}X of finite
index. Then ΩX = ℛ_{σ}X for sufficiently large σ, and the following result is
proved:
THEOREM C. If G is an abelian pgroup of Ulm type τ such that ΩA(G) = 1, then
ℛ_{τ+1}A(G) = 1. Moreover, if G is reduced, then ℛ_{τ}A(G) = 1.
