## Characters of finite quasigroups. VII: Permutation characters.(English)Zbl 1101.20037

Summary: Each homogeneous space of a quasigroup affords a representation of the Bose-Mesner algebra of the association scheme given by the action of the multiplication group. The homogeneous space is said to be faithful if the corresponding representation of the Bose-Mesner algebra is faithful. In the group case, this definition agrees with the usual concept of faithfulness for transitive permutation representations. A permutation character is associated with each quasigroup permutation representation, and specialises appropriately for groups. However, in the quasigroup case the character of the homogeneous space determined by a subquasigroup need not be obtained by induction from the trivial character on the subquasigroup. The number of orbits in a quasigroup permutation representation is shown to be equal to the multiplicity with which its character includes the trivial character.
For Part VI of this series cf. [Eur. J. Comb. 11, No. 3, 267–275 (1990; Zbl 0704.20056)].

### MSC:

 20N05 Loops, quasigroups 05E30 Association schemes, strongly regular graphs 20C99 Representation theory of groups

Zbl 0704.20056
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