The axiomatic closure of the class of discriminating groups.

*(English)*Zbl 1080.20025A 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.

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.

Reviewer: Oleg V. Belegradek (Istanbul)

##### 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 |