The hypo residuum of the automorphism group of an abelian \(p\)-group (Q2537784)

The hypo residuum \(\Omega X\) of a group \(X\) is defined as the product of all normal subgroups \(N\) of \(X\) such that 1 is the only finite epimorphic image of \(N\). Let \(G\) be an abelian \(p\)-group and \(A(G)\) its automorphism group. It is shown that \(\Omega A(G)=1\) if and only if every divisible and every bounded pure subgroup of \(G\) has finite rank. Equivalent are: (i) \(G\) possesses a quasi-cyclic group of automorphisms; (ii) every countable group is a group of automorphisms of \(G\); (iii) the group of all permutations on a countably infinite set can be embedded into \(A(G)\); (iv) \(\Omega A(G)\neq 1\). Let \(\Gamma(G)\) denote the set of all automorphisms \(\gamma\) of \(G\) such that the height of \(g\gamma-g\) is infinite for every \(g\in G\). It is shown that, whenever \(G\) is reduced (or the maximal divisible subgroup of \(G\) has finite rank), then \(\Omega\Gamma(G)=1\), and furthermore, that in this case \(\Omega A(G)=1\) if and only if \(A(G)/\Gamma(G)\) is residually finite.