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

2007/01/25 by Vladimir Tolstykh, Tolstykh, Vladimir · 1 citation
Mathematics · #03C60 (20F28 #20K30) #Advanced Differential Equations and Dynamical Systems #Advanced Topology and Set Theory #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.math/0701752

openalex publication_date 2007/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 A₁,A₂ are elementarily equivalent if and only if the sets |A₁|,|A₂| are second-order equivalent.

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