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Oleg Belegradek

Discriminating and square-like groups

A group is discriminating if and only if it discriminates its direct square, and square-like if and only if it is universally equivalent to its direct square. It is known that any discriminating group is square-like. These notions were introduced and studied in a series of papers by Baumslag, Myasnikov and Remeslennikov and by Fine, Gaglione, Myasnikov and Spellman. We prove that any square-like group is elementarily equivalent to a countable discriminating group. This answers a question of the second group of authors. We provide an explicit universal–existential axiom system for the class of square-like groups. We show that the theory of the class of discriminating groups is computably enumerable but undecidable. We give a criterion for determining whether a group is discriminating. We propose a construction method for discriminating groups and use it to construct in various group varieties many discriminating non-abelian groups that do not embed their squares. We construct square-like, non-discriminating nilpotent p-groups of arbitrary nilpotency class; all previously known square-like, non-discriminating groups were abelian.

Journal of Group Theory, Walter de Gruyter

Print ISSN: 1433-5883
Volume: 7, 09/2004
Pages: 521 - 532

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