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Journal of the London Mathematical Society 2002 66(2):325-333; doi:10.1112/S0024610702003484
© 2002 by London Mathematical Society
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© The London Mathematical Society

Transitive Permutation Groups Without Semiregular Subgroups

Peter J. Cameron, Michael Giudici, Gareth A. Jones, William M. Kantor, Mikhail H. Klin, Dragan Marusic and Lewis A. Nowitz

School of Mathematical Sciences, Queen Mary, University of London Mile End Road, London E1 4NS
Department of Mathematics, University of Southampton Southampton SO17 1BJ
Department of Mathematics, University of Oregon Eugene, OR 97403, USA
Department of Mathematics, Ben-Gurion University of the Negev PO Box 653, 84105 Beer-Sheva, Israel
2345 Broadway 526, New York, NY 10024-3213, USA
IMFM, Oddelek za Matematiko, Univerza v Ljubljani Jadranska 19, 1000 Ljubljana, Slovenia

Received 4 April 2001. Revision received 14 January 2002.

A transitive finite permutation group is called elusive if it contains no nontrivial semiregular subgroup. The purpose of the paper is to collect known information about elusive groups. The main results are recursive constructions of elusive permutation groups, using various product operations and affine group constructions. A brief historical introduction and a survey of known elusive groups are also included. In a sequel, Giudici has determined all the quasiprimitive elusive groups.

Part of the motivation for studying this class of groups was a conjecture due to Marusic, Jordan and Klin asserting that there is no elusive 2-closed permutation group. It is shown that the constructions given will not build counterexamples to this conjecture.


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