Vol. 35, No. 1, 1970

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The hypo residuum of the automorphism group of an abelian p-group

Jutta Hausen

Vol. 35 (1970), No. 1, 127–139

Throughout the following G denotes an abelian p-group (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 P-group if and only if A(G) is a Q-group, where P and Q are group theoretical properties, have been proved by R. Baer; some others are well-known.

This paper gives a characterisation of this kind for the class P of all abelian p-groups whose homogeneous direct summands have finite rank (an abelian p-group 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 l-automorphism in GlpωG is of special significance. It is shown that ΩΓ(G) = 1 for every reduced abelian p-group 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 μX of X: let 0X = X, define λX = μ<λμX if λ is a limit ordinal and μX has been defined already for all ordinals μ < λ, and let μ+1X be the intersection of all subgroups of μX of finite index. Then ΩX = σX for sufficiently large σ, and the following result is proved:

THEOREM C. If G is an abelian p-group of Ulm type τ such that ΩA(G) = 1, then τ+1A(G) = 1. Moreover, if G is reduced, then τA(G) = 1.

Mathematical Subject Classification
Primary: 20.22
Received: 23 April 1969
Revised: 18 February 1970
Published: 1 October 1970
Jutta Hausen