Sharp Characters of Quasigroups

The idea of a sharp permutation character of a group arises from combinatorial considerations. Recent work, founded on early results of Blichfeldt, has shown that the definition of sharpness can be extended to arbitrary group characters, and in fact it has emerged that the natural object to define is a sharp triple. In the present work the definition is extended further to quasigroup characters, which have been defined and discussed in [7-12]. In the more general context subtleties arise because the coefficient ring which is taken in relation to character products is no longer Z. Examples are given here of sharp characters and triples which come from non-associative loops and quasigroups in various ways. Results are presented on sharp characters which take on a small number of values. It is also pointed out that all irreducible triples arising from an abelian group are sharp.