Discriminating groups were introduced in  with an eye toward applications to the universal theory of various groups. In  it was shown that if G is any discriminating group, then the universal theory of G coincides with that of its direct square G x G. In this paper we explore groups G whose universal theory coincides with that of their direct square. These are called square-like groups. We show that the class of square-like groups is first-order axiomatizable and contains the class of discriminating groups as a proper subclass. Further we show that the class of discriminating groups is not first-order axiomatizable.
Journal of Group Theory
Fine, Benjamin; Gaglione, Anthony M.; Myasnikov, A. G.; and Spellman, Dennis, "Groups Whose Universal Theory Is Axiomatizable by Quasi-Identities" (2002). Math & Computer Science Faculty Publications. 6.
B. Fine, A. M. Gaglione, A. G. Myasnikov, and D. Spellman. Groups Whose Universal Theory Is Axiomatizable by Quasi-Identities, Journal of Group Theory. 5(3), 365-381.