## K.W. Johnson, J.D.H. Smith

*Characters of finite quasigroups VII: permutation characters *

Comment.Math.Univ.Carolinae 45,2 (2004) 265-273. **Abstract:**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.

**Keywords:** quasigroup, association scheme, permutation character

**AMS Subject Classification:** 20N05

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