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

2005/01/01 by Vladimir Tolstykh · 1 citation
Mathematics · Chemistry · #Advanced Topology and Set Theory #Geometric and Algebraic Topology #Rings, Modules, and Algebras #Mathematics #Abelian group #Alternating group #Outer automorphism group #Group (periodic table) #Inner automorphism #G-module #Automorphism #p-group #Automorphism group #Free group #Non-abelian group #Combinatorics #Elementary abelian group #Free abelian group #Pure mathematics #Rank of an abelian group #Chemistry

paper · doi:10.1090/conm/380/07117

openalex publication_date 2005/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/03/12

Abstract

Let A be an infinitely generated free abelian group.We prove that the automorphism group Aut(A) first-order interprets the full secondorder 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.

Cited by

• Free abelian group [wikipedia]