Belyavskaya, G. B. Identities with permutations and quasigroups isotopic to groups and to Abelian groups. (English. Russian original) Zbl 1315.20060 Discrete Math. Appl. 23, No. 3-4, 369-384 (2013); translation from Diskretn. Mat. 25, No. 2, 68-81 (2013). Summary: This paper continues the study of quasigroups isotopic to groups and to Abelian groups. For identities with permutations of a certain kind the notion of the type is introduced; identities with permutations of various types are presented, which characterize quasigroups isotopic to Abelian groups. The approach being used had allowed to establish a number of new unbalanced identities involving four variables in an equasigroup, as well as a number of identities involving three variables in an equasigroup with one nullary operation, each of which characterises a quasigroup isotopic to a group or to an Abelian group. MSC: 20N05 Loops, quasigroups Keywords:quasigroups isotopic to groups; quasigroups isotopic to Abelian groups; parastrophic equivalence of identities; identities with permutations; balanced identities PDFBibTeX XMLCite \textit{G. B. Belyavskaya}, Discrete Math. Appl. 23, No. 3--4, 369--384 (2013; Zbl 1315.20060); translation from Diskretn. Mat. 25, No. 2, 68--81 (2013) Full Text: DOI