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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].

##### MSC:
 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