On Bruck-Belousov Problem

In this paper on the language of subgroups of the multiplication group of a quasigroup (of the associated group of a quasigroup) necessary and sufficient conditions of normality of congruences of a left (right) loop are given. These conditions can be considered as a partial answer to the problem posed in books of R. H. Bruck and V. D. Belousov about conditions of normality of all congruences of quasigroups. Results on the regularity of congruences of quasigroups and the behavior of quasigroup congruences by isotopy are given.