Quasigroups satisfying balanced but not Belousov equations are group isotopes

Abstract

V. D. Belousov (1925-88) made numerous contributions to the study of quasigroups. In particular, his lengthy 1966 paper "Balanced identities in quasigroups" [4] contains what has been described as a "very significant" and "remarkable" theorem [11, pp. 68-69]. Remarkable though it was, this theorem provided only a partial answer to the question as to which balanced equations on quasigroups gave rise to group isotopes. Although not specifically addressed in the paper [12], a characterization of the balanced equations in question may be derived from a generalization of Belousov's Theorem due to E. Falconer. The first author explicitly solved the problem in 1979; however his characterization was of a technical nature and depended on machinery developed over three papers [13]. In 1985 Belousov found a characterization which is not only elegant but also lends itself to a simple proof [5]. The purpose of this paper is to provide sufficient background for the non specialist to understand and enjoy what we too would describe as "a remarkable theorem".