Bounds on finite quasiprimitive permutation groups

Abstract

A permutation group is said to be quasiprimitive if every nontrivial normal subgroup is transitive. Every primitive permutation group is quasiprimitive, but the converse is not true. In this paper we start a project whose goal is to check which of the classical results on finite primitive permutation groups also holds for quasiprimitive ones (possibly with some modifications). The main topics addressed here are bounds on order, minimum degree and base size, as well as groups containing special p -elements. We also pose some problems for further research.2000 Mathematics subject classification: primary 20B05, 20B15; secondary 20B35.

Bounds and quotient actions of innately transitive groups

“…https://doi.org/10.1017/S1446788700009356 group containing special permutations, a bound on the base size of a quasiprimitive group, and a bound on the minimal degree of a quasiprimitive group. Praeger and Shalev also give a nice exposition of the literature in [10], which we do not repeat here.…”

“…In Section 5, we give an expository summary of the definitions and elementary properties of permutational transformations. Our goal in Sections 5-8 is to state and prove a theorem which encapsulates seven generalisations of results from [10]. Finally, in Section 9, we give a complete account of the innately transitive types of quotient actions of innately transitive groups, in a similar manner to that of Praeger's [7] investigation of quotient actions of quasiprimitive groups.…”

A theory of semiprimitive groups

“…A well-known result due to Jordan states that a primitive permutation group of degree n containing a cycle of prime length p with p n − 3 must contain Alt(n). This result was extended to finite quasiprimitive and finite innately transitive groups in [30] and [4] respectively. Here we show that the same result holds in the context of semiprimitive groups (of arbitrary cardinality).…”

“…This leads to a natural generalisation where we call a permutation group quasiprimitive if each of its non-trivial normal subgroups is transitive. Many questions about permutation groups can be reduced to questions about primitive or quasiprimitive groups and they have been the focus of much attention, for example [1,8,9,10,13,14,16,17,19,23,25,26,27,28,30,31,32].Innately transitive permutation groups were introduced by Bamberg and Praeger [3] and these are the finite permutation groups G with a transitive minimal normal subgroup N. Such groups naturally occur as overgroups of quasiprimitive groups. A permutation group G is called semiregular if each point-stabiliser G ω is trivial.…”

“…Thus the density of the set Deg sp is at least 1 2 . We are thus lead to the following: 30 Problem 2a: What is the density of the set Deg sp ? Is it less than 1?…”