Elementary equivalence of the automorphism groups of Abelian <Emphasis Type="Italic">p</Emphasis>-groups

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.