What does the automorphism group of a free abelian group A know about A?
Abstract
Let A be an infinitely generated free abelian group. We prove that the automorphism group 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 A1,A2 are elementarily equivalent if and only if the sets |A1|,|A2| are second-order equivalent.
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