On generalized multiplication groups of the commutative Moufang loops

Abstract

The multiplication groups of quasigroups, i.e. the groups generated by all left and right translations, represent an efficient tool in the theory of quasigroups (loops). Belousov considered the groups, generated by all left, right and middle translations of a quasigroup, called the generalized multiplication group. He remarked that these groups are invariant under parastrophy of quasigroups, and found a set of generators for the stabilizer of a fixed element in the generalized multiplication group. The generalized multiplication groups and the generalized inner mapping groups are invariant under the isostrophy of loops.