### 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 char- acter is associated with each quasigroup permutation representation, and specialises appropriately for groups. However, in the quasigroup case the character of the homoge- neous 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 permu- tation representation is shown to be equal to the multiplicity with which its character includes the trivial character.

Original language | English (US) |
---|---|

Pages (from-to) | 265-273 |

Number of pages | 9 |

Journal | Commentationes Mathematicae Universitatis Carolinae |

Volume | 45 |

Issue number | 2 |

State | Published - Jan 1 2004 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Commentationes Mathematicae Universitatis Carolinae*,

*45*(2), 265-273.

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*Commentationes Mathematicae Universitatis Carolinae*, vol. 45, no. 2, pp. 265-273.

**Characters of finite quasigroups VII : Permutation characters.** / Johnson, Kenneth; Smith, J. D.H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Characters of finite quasigroups VII

T2 - Permutation characters

AU - Johnson, Kenneth

AU - Smith, J. D.H.

PY - 2004/1/1

Y1 - 2004/1/1

N2 - 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 char- acter is associated with each quasigroup permutation representation, and specialises appropriately for groups. However, in the quasigroup case the character of the homoge- neous 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 permu- tation representation is shown to be equal to the multiplicity with which its character includes the trivial character.

AB - 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 char- acter is associated with each quasigroup permutation representation, and specialises appropriately for groups. However, in the quasigroup case the character of the homoge- neous 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 permu- tation representation is shown to be equal to the multiplicity with which its character includes the trivial character.

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M3 - Article

AN - SCOPUS:33846798647

VL - 45

SP - 265

EP - 273

JO - Commentationes Mathematicae Universitatis Carolinae

JF - Commentationes Mathematicae Universitatis Carolinae

SN - 0010-2628

IS - 2

ER -