Quasigroup actions: Markov chains, pseudoinverses, and linear representations. (English) Zbl 0944.20059

This paper initiates the extension to quasigroups of a further aspect of group representation theory: transitive permutation representations. Using pseudoinverses of incidence matrices of quasigroups in partitions induced by left multiplications of subquasigroups, a transitive permutation action of a quasigroup is defined as a set of Markov chain actions indexed by the quasigroup. Finally, Burnside’s lemma for transitive quasigroup actions is derived.


20N05 Loops, quasigroups
20C15 Ordinary representations and characters
20B05 General theory for finite permutation groups
20M30 Representation of semigroups; actions of semigroups on sets