AUTOMORPHISMS OF BOUNDED ABELIAN-GROUPS

We show that if A is a countable abelian group of finite exponent, and H is a subgroup of index less than 2-omega in the automorphism group Aut (A) of A, then H contains the pointwise stabiliser of some finite set of elements of A. (Thus A has the "small index property.") We consider groups B which are direct sums of isomorphic cyclic p-groups Z (p(m)), together with subgroups A which are the corresponding sums of p-groups Z(p(l)) for some l < m. For these pairs of groups we determine exactly when it is true that each automorphism of A extends to an automorphism of B, so that the extensions form an embedding of Aut(A) in Aut(B). Finally we discuss several connections between these results and questions in set theory and model theory.