What does the automorphism group of a free abelian group A know about A?

Vladimir Tolstykh

paper

What does the automorphism group of a free abelian group A know about A?

arXiv:math/0701752

Abstract

Let $A$ be an infinitely generated free abelian group. We prove that the automorphism group $\aut A$ first-order interprets the full second-order theory of the set $∣ A ∣$ with no structure. In particular, this implies that the automorphism groups of two infinitely generated free abelian groups $A_{1}, A_{2}$ are elementarily equivalent if and only if the sets $∣ A_{1} ∣, ∣ A_{2} ∣$ are second-order equivalent.

A pre-publication preprint of a paper published in `2005