Elementary equivalence of the automorphism groups of Abelian p-groups

Abstract

We consider Abelian p-groups (p ≥ 3) A 1 = D 1 ⊕ G 1 and A 2 = D 2 ⊕ G 2, where D 1 and D 2 are divisible and G 1 and G 2 are reduced subgroups. We prove that if the automorphism groups Aut A 1 and Aut A 2 are elementarily equivalent, then the groups D 1, D 2 and G 1, G 2 are equivalent, respectively, in the second-order logic.