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On connected transversals to Abelian subgroups

Published online by Cambridge University Press:  17 April 2009

Markku Niemenmaa  and
Tomas Kepka
Markku Niemenmaa
Affiliation:
Department of Mathematics, University of Oulu, SF-90570 Oulu, Finland
Tomas Kepka
Affiliation:
MFF - UK Prague, Czechoslovakia
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Abstract

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In this paper we investigate the situation where a group G has an abelian subgroup H with connected transversals. We show that if H is finite then G is solvable. We also investigate some special cases where the structure of H is very close to the structure of a cyclic group. Finally we apply our results to loop theory and we show that if the inner mapping group of a finite loop Q is abelian then Q is centrally nilpotent.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 49 , Issue 1 , February 1994 , pp. 121 - 128
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Bruck, R.H., ‘Contributions to the theory of loops’, Trans. Amer. Math. Soc. 60 (1946), 245–354.CrossRefGoogle Scholar
[2]Conway, J.H., ‘A simple construction for the Fischer-Griess monster group’, Invent. Math. 79 (1985), 513–540.Google Scholar
[3]Glauberman, G., ‘On loops of odd order’, J. Algebra 1 (1964), 374–396.Google Scholar
[4]Glauberman, G., ‘On loops of odd order II’, J. Algebra 8 (1968), 393–414.CrossRefGoogle Scholar
[5]Griess, R.L., ‘A Moufang loop, the exceptional Jordan algebra and a cubic form in 27 variables’, J. Algebra 131 (1990), 281–293.Google Scholar
[6]Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin, Heidelberg, New York, 1967).Google Scholar
[7]Kepka, T. and Niemenmaa, M., ‘On conjugacy classes in finite loops’, Bull. Austral. Math. Soc. 38 (1988), 171–176.Google Scholar
[8]Liebeck, M.W., ‘The classification of finite simple Moufang loops’, Math. Proc. Camb. Phil. Soc. 102 (1987), 33–47.CrossRefGoogle Scholar
[9]Niemenmaa, M., ‘Problems in loop theory for group theorists’, in Groups St.Andrews 1989 Vol.2, London Math. Soc. Lecture Notes Series 160 (Cambridge University Press, Cambridge, 1991), pp. 396–399.Google Scholar
[10]Niemenmaa, M. and Kepka, T., ‘On multiplication groups of loops’, J. Algebra 135 (1990), 112–122.CrossRefGoogle Scholar
[11]Niemenmaa, M. and Kepka, T., ‘On connected transversals to abelian subgroups in finite groups’, Bull. London Math. Soc. 24 (1992), 343–346.CrossRefGoogle Scholar
[12]Niemenmaa, M. and Kepka, T., ‘On loops with cyclic inner mapping groups’, Arch. Math. (to appear).Google Scholar