Journal of the London Mathematical Society



Notes and Papers

TRANSITIVE PERMUTATION GROUPS WITHOUT SEMIREGULAR SUBGROUPS


PETER J. CAMERON a1, MICHAEL GIUDICI a1, GARETH A. JONES a2, WILLIAM M. KANTOR a3 1 , MIKHAIL H. KLIN a4, DRAGAN MARUŠIC a6 and LEWIS A. NOWITZ a5
a1 School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS
a2 Department of Mathematics, University of Southampton, Southampton SO17 1BJ
a3 Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
a4 Department of Mathematics, Ben-Gurion University of the Negev, PO Box 653, 84105 Beer-Sheva, Israel
a5 2345 Broadway 526, New York, NY 10024-3213, USA
a6 IMFM, Oddelek za Matematiko, Univerza v Ljubljani, Jadranska 19, 1000 Ljubljana, Slovenia

Abstract

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 Marušic, 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.

(Received April 4 2001)
(Revised January 14 2002)



Footnotes

1 The fourth author was supported in part by the National Science Foundation.