Characters of finite quasigroups. V: Linear characters. (English) Zbl 0679.20059

[Part IV, cf. ibid. 10, No.3, 257-263 (1989; Zbl 0669.20053.]
Aspects of the structure of quasigroups with a unique non-linear basic character are investigated. For example, in Section 4 it is shown that the linear basic characters of a general quasigroup are well behaved under multiplication. They form an abelian group which then acts on the remaining basic characters. This action extends to give an abelian group of isometries of the space of class functions. In the final Section the authors examine finite, non-empty quasigroups having a unique non-linear basic character.
Reviewer: C.Pereira da Silva


20N05 Loops, quasigroups
20C99 Representation theory of groups


Zbl 0669.20053
Full Text: DOI


[1] Bannai, E.; Ito, T., Algebraic combinatorics I, (1984), Benjamin/Cummings Menlo Park, California · Zbl 0555.05019
[2] ()
[3] Conway, J.H., A simple construction for the fischer-Griess monster group, Inv. math., 79, 513-540, (1985) · Zbl 0564.20010
[4] Johnson, K.W.; Smith, J.D. H., Characters of finite quasigroups, Europ. J. combinatorics, 5, 43-50, (1984) · Zbl 0537.20042
[5] Johnson, K.W.; Smith, J.D. H., Characters of finite quasigroups II: induced characters, Europ. J. combinatorics, 7, 131-137, (1986) · Zbl 0599.20110
[6] Johnson, K.W.; Smith, J.D. H., Characters of finite quasigroups III: quotients and fusion, Europ. J. combinatorics, 10, 47-56, (1989) · Zbl 0667.20053
[7] Johnson, K.W.; Smith, J.D. H., Characters of finite quasigroups IV: products and superschemes, Europ. J. combinatorics, 10, 257-263, (1989) · Zbl 0669.20053
[8] Romanowska, A.B.; Smith, J.D. H., Modal theory, (1985), Heldermann Berlin · Zbl 0553.08001
[9] Smith, J.D. H., Centraliser rings of multiplication groups on quasigroups, Math. proc. camb. phil. soc., 79, 427-431, (1976) · Zbl 0335.20035
[10] Smith, J.D. H., Representation theory of infinite groups and finite quasigroups, (1986), University de Montreal Montreal · Zbl 0609.20042
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.