## 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.

### MSC:

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