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The axiomatic closure of the class of discriminating groups. (English) Zbl 1080.20025
A group $$G$$ is called discriminating if every group separated by $$G$$ is discriminated by $$G$$. Here $$G$$ is said to separate (discriminate) a group $$H$$ if for any non-identity element (finite set of non-identity elements) of $$H$$ there is a homomorphism from $$H$$ to $$G$$ which does not map the element (each element of the set) to the identity.
This notion was introduced by G. Baumslag, A. Myasnikov, and V. Remeslennikov [in J. Group Theory 3, No. 4, 467-479 (2000; Zbl 0965.20014)], and studied by the authors in a series of papers. It was shown that $$G$$ is discriminating iff $$G$$ discriminates $$G^2$$, and that if $$G$$ is discriminating then the groups $$G$$ and $$G^2$$ are universally equivalent. The authors call groups $$G$$ with the latter property square-like. In [Combinatorial and geometric group theory, Contemp. Math. 296, 129-137 (2002; Zbl 1017.20025)] they proved that any Abelian square-like group is elementarily equivalent to a discriminating group; the question whether this is true for arbitrary square-like groups remained open.
The main result of the paper under review is a positive solution to the problem. A consequence is that the class of square-like groups is the axiomatic closure of the class of discriminating groups. (Note that the reviewer [J. Group Theory 7, No. 4, 521-532 (2004; Zbl 1077.20051)] independently proved a bit stronger result: a group is square-like iff it is elementarily equivalent to a countable group that embeds its square.) Also, it is derived from a result of F. Oger’s that for every $$n$$ there are square-like groups $$G$$ and $$H$$ such that $$G\equiv_nH$$ but not $$G\equiv H$$. Another observation in the paper is that the notion of discriminating group considered here is stronger than another notion of discriminating group previously known for group varieties.

MSC:
 20E26 Residual properties and generalizations; residually finite groups 03C60 Model-theoretic algebra 20A15 Applications of logic to group theory 20E10 Quasivarieties and varieties of groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F05 Generators, relations, and presentations of groups
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