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On conjugacy classes in finite loops

Published online by Cambridge University Press:  17 April 2009

T. Kepka  and
M. Niemenmaa
T. Kepka
Affiliation:
Charles University, Prague, Czechoslovakia
M. Niemenmaa
Affiliation:
Department of Mathematics, University of Oulu, Linnanmaa, 90570 Oulu, Finland
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Abstract

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The rôle of the conjugacy relation is certainly important in the structure theory of groups. Here we study this relation in a considerably more general setting, namely in the theory of loops. We first recall some basic facts about quasigroups, their multiplication groups, their inner mapping groups and the conjugacy relation. After this we estimate the size and the number of the conjugacy classes and we study the structure of loops having only two conjugacy classes. Finally, the values of the centraliser function are discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 38 , Issue 2 , October 1988 , pp. 171 - 176
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Bertram, E.A., ‘Large centralizers in finite soluble groups’, Israel J. Math. 47 (1984), 335–344.CrossRefGoogle Scholar
[2]Brauer, R. and Fowler, K., ‘On groups of even order’, Ann. Math. 82 (1955), 565–583.CrossRefGoogle Scholar
[3]Bruck, R., A Survey of Binary Systems (Springer-Verlag, Berlin—Heidelberg—New York, 1971).CrossRefGoogle Scholar
[4]Cossey, J., ‘Finite soluble groups have large centralizers’, Bull. Austral. Math. Soc. 35 (1987), 291–298.CrossRefGoogle Scholar
[5]Dénes, J. and Keedwell, A.D., Latin Squares and Their Applications (Akadémiai Kiadó, Budapest, 1976).Google Scholar
[6]Drápal, A. and Kepka, T., ‘Alternating groups and latin squares’ (to appear).Google Scholar
[7]Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin—Heidelberg—New York, 1967).CrossRefGoogle Scholar
[8]Ihringer, T., ‘Quasigroups, loops and centralizer rings’, in Contributions to General Algebra 3.: Proceedings of the Vienna Conference, pp. 211–224, 1984.Google Scholar
[9]Newman, M., ‘A bound for the number of conjugacy classes in a group’, J. London Math. Soc. 43 (1968), 108–110.CrossRefGoogle Scholar
[10]Smith, J.D.H., ‘Multiplication Groups of Quasigroups, Preprint 603, Darmstadt, 1981.Google Scholar