Transitive Permutation Groups Without Semiregular Subgroups

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 Marušič, 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.

Characterizing a family of elusive permutation groups

“…The automorphism group of a digraph is 2-closed, and Marusič [12] originally asked if every vertex transitive digraph has a fixed-point free automorphism of prime order. This was later extended to 2-closed groups by Klin [3] and has recently received a lot of attention, for example in [2], [4], [5], [8], [9], [13]. It was shown in [7, p. 2723] that the elusive permutation groups in Theorem 1.1 are not 2-closed and that their 2-closures are not elusive.…”

“…Note that N has p þ 1 orbits of length p. The elusive groups provided by the doubling construction in [2,Theorem 3.3] using an FKS-group as input also have this abstract structure. The doubling con-struction was generalized in [7] (see Construction 2.3) to a priming construction which multiplies the order of G 1 by a prime dividing p À 1.…”

“…These groups are of the form G ¼ N z G 1 where N G C d p and G 1 is a 2-subgroup of the multiplicative group of GFðp d Þ. It was shown in [10] that when p ¼ 3 this does in fact produce elusive permutation groups while the doubling construction in [2] shows that it produces examples for any Mersenne prime p. Theorem 1.1 shows that these are the only primes for which the construction produces elusive groups.…”

“…It was also shown in [8] that the only elusive groups with a transitive minimal normal subgroup are the groups M 11 wr K acting on 12 k points with K a transitive subgroup of S k . More examples and constructions of elusive groups were given in [2], [7].…”

“…We say that G is 2-closed if it is equal to its 2-closure. The polycirculant conjecture (see [2]) states that every 2-closed transitive permutation group has a fixed-point free element of prime order. The automorphism group of a digraph is 2-closed, and Marusič [12] originally asked if every vertex transitive digraph has a fixed-point free automorphism of prime order.…”