Belegradek, O2024-07-182024-07-1820041433-58831435-4446https://doi.org/10.1515/jgth.2004.7.4.521https://hdl.handle.net/11411/8094A 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, nondiscriminating nilpotent p-groups of arbitrary nilpotency class; all previously known square-like, non-discriminating groups were abelian.eninfo:eu-repo/semantics/closedAccessDiscriminating and square-like groupsArticle2-s2.0-634423013110.1515/jgth.2004.7.4.5215324Q25217Q2WOS:000223773200007