Characters of finite quasigroups. (English) Zbl 0537.20042

In this paper the authors study some of the basics of a character theory for finite non-empty quasigroups, generalizing the traditional ordinary character theory for groups. The authors present the relevant notions from quasigroup theory (the multiplication group, quasigroup conjugacy classes, and the space of class functions) which enable one to apply these theories to quasigroups. Quasigroup characters as class functions and the character table of a quasigroup are given explicitly, and the characters are seen to form an orthonormal basis for the space of class functions. Main result: the character table of a finite quasigroup Q specifies the congruence lattice of Q.
Reviewer: C.Pereira da Silva


20N05 Loops, quasigroups
Full Text: DOI


[1] Baer, R., Nets and groups I, Trans. amer. math. soc., 46, 110-141, (1939) · JFM 65.0819.02
[2] Brauer, R., On pseudo groups, J. math. soc. Japan, 20, 13-22, (1968) · Zbl 0185.06503
[3] Bruck, R.H., Contributions to the theory of loops, Trans. amer. math. soc., 60, 245-354, (1946) · Zbl 0061.02201
[4] Bruck, R.H., A survey of binary systems, (1966), Springer-Verlag Berlin · Zbl 0141.01401
[5] Cameron, P.J., Suborbits in transitive permutation groups, (), 419-450 · Zbl 0297.20003
[6] Cameron, P.J.; Goethals, J.M.; Seidel, J.J., The Krein condition, spherical designs, norton algebras and permutation groups, Indag. math., 81, 196-206, (1978) · Zbl 0399.20001
[7] Cohn, P.M., Universal algebra, (1965), Harper and Row New York
[8] Cohn, P.M., Algebra vol. 1, (1974), John Wiley Chichester, Sussex
[9] Curtis, C.W.; Reiner, I., Representation theory.of finite groups and associative algebras, (1962), Interscience New York
[10] Delsarte, P., An algebraic approach to the association schemes of coding theory, Philips res. repts., Suppl. 10, (1973) · Zbl 1075.05606
[11] Harrison, D.K., Double coset and orbit spaces, Pac. J. math., 80, 451-491, (1979) · Zbl 0415.20022
[12] Higman, D.G., Coherent configurations part I, Geometriae dedicata, 4, 1-32, (1975) · Zbl 0333.05010
[13] Isaacs, I.M., Character theory of finite groups, (1976), Academic Press New York · Zbl 0337.20005
[14] Johnson, K.W., Transversals, S-rings and centraliser rings of groups, (), Springer Lecture Notes in Mathematics 848 · Zbl 0463.20004
[15] Johnson, K.W., Loop transversals and the centraliser ring of a permutation group, Math. proc. camb. phil. soc., 94, 411-416, (1983) · Zbl 0597.20002
[16] Neumann, P.M., Finite permutation groups, edge coloured graphs and matrices, () · Zbl 0382.20003
[17] Smith, J.D.H., Centraliser rings of multiplication groups on quasigroups, Math. proc. camb. phil. soc., 79, 427-431, (1976) · Zbl 0335.20035
[18] Smith, J.D.H., Mal’cev varieties, Springer lecture notes in mathematics 554, (1976), Springer-Verlag Berlin · Zbl 0344.08002
[19] Tamaschke, O., S-ringe and verallgemeinerte charaktere auf endlichen gruppen, Math. zeitschr., 84, 101-119, (1964) · Zbl 0126.05801
[20] Tamaschke, O., On Schur-rings which define a proper character theory on finite groups, Math. Z., 117, 340-360, (1970) · Zbl 0204.35203
[21] Wieland, H., Finite permutation groups, (1964), Academic Press New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.