Elementary equivalence of the automorphism groups of Abelian p-groups

Abstract

We prove that two elementarily equivalent Abelian p-groups with separable reduced parts are isotypically equivalent. Also as the corollary we prive that any Abelian p-group with a separable reduced part if ω-strongly homogeneous.

A Criterion of Elementary Equivalence of Automorphism Groups of Unreduced Abelian p-Groups

“…The |B|-bounded second-order theory of the group G is expressed by the first-order language of Aut A in [18]. The second-order theory of the divisible part D is expressed in [9]. Hence, expressibility of the second-order theory of the entire group A is already proved for the case |G| = |B|.…”

“…In [9], E. I. Bunina and M. A. Roizner introduced certain formulas for operating with involutions (i.e., automorphisms of order 2). The declarations of these formulas follow below.…”

“…The second-order theory of the group B is also expressed in [9] We begin with defining an arbitrary quasicyclic summand…”

“…Then there is a subgroup in D that can be identified with G/B by an isomorphism h ∈ Hom(G/B, D). In [9], it is shown how one can express the second-order theories of the groups D and B in the group D in the case where |D| ≥ |B|. Since G/B is identified with a subgroup of D, one can express the second-order theories of the groups D, B, and G/B in D. Thus, one can express the second-order theory of the entire group A in D. The description of this interpretation follows.…”

“…Details of the interpretation of second-order logic and of precise formulas of "addition" and "equality" of encoding automorphisms are described in [9]. Now let |G/B| > |D|.…”