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Every Abelian group universally equivalent to a discriminating group is elementarily equivalent to a discriminating group. (English) Zbl 1017.20025
Cleary, Sean (ed.) et al., Combinatorial and geometric group theory. Proceedings of the AMS special session on combinatorial group theory, New York, NY, USA, November 4-5, 2000 and the AMS special session on computational group theory, Hoboken, NJ, USA, April 28-29, 2001. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 296, 129-137 (2002).
A group \(G\) is discriminating if for every finite nonempty subset \(S\) of \(G\times G\) not containing \(1\times 1\) there is a homomorphism \(\varphi_S\colon G\to G\) such that \(\varphi_S(g)\neq 1\) for all \(g\in S\). A group \(H\) is squarelike if there is a discriminating group \(G_H\) universally equivalent to \(H\) (that is, they satisfy the same universal sentences of \(L\)). Two groups are elementarily equivalent if they satisfy precisely the same sentences of \(L\). The main results of the paper provide partial answers to questions raised in an earlier paper.
Theorem 2.1. Every squarelike torsion Abelian group is the direct union of a family of discriminating subgroups.
Corollary 2.7. Every squarelike Abelian group is elementarily equivalent to a discriminating group.
For the entire collection see [Zbl 0990.00044].

20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20K99 Abelian groups
08C10 Axiomatic model classes
20E10 Quasivarieties and varieties of groups
03C60 Model-theoretic algebra
20A15 Applications of logic to group theory