Permutation representations of loops.
The author initiates a theory of permutation representations of finite loops. Such a theory can be introduced more generally for a finite quasigroup \(Q\) by considering, for a given subquasigroup \(P\), the elements of the corresponding homogeneous space \(P\setminus Q\) as orbits on \(Q\) of the group of permutations generated by the left multiplications by elements of \(P\). Each element of \(Q\) yields a Markov chain action on the homogeneous space \(P\setminus Q\) as a set of states. If \(P\) is a subgroup of a group \(Q\) this concept specializes to the usual one of a homogeneous space or transitive permutation representation for groups. For finite loops the author obtains that the class of permutation representations is closed under disjoint unions and direct products, each representation decomposing into a disjoint union of irreducible representations. But, in contrast with the group case, he shows that a loop need not to be recoverable to within isomorphism from a faithful permutation representation. The final section is devoted to an interesting application of this theory to the study of the so-called `Lagrange properties' of a finite loop.
scientific article; zbMATH DE number 1956738
scientific article