A coalgebraic approach to quasigroup permutation representations

Abstract

The paper identifies the class of all permutation representations of a given finite quasigroup as a covariety of coalgebras. Each permutation representation decomposes as a sum of homomorphic images of homogeneous spaces. For a group, permutation representations in the present sense specialise to the classical concept. Burnside's Lemma, with a new proof, is extended from groups to quasigroups.