Permutation Groups and Cartesian Decompositions

Abstract

Permutation groups, their fundamental theory and applications are discussed in this introductory book. It focuses on those groups that are most useful for studying symmetric structures such as graphs, codes and designs. Modern treatments of the O'Nan–Scott theory are presented not only for primitive permutation groups but also for the larger families of quasiprimitive and innately transitive groups, including several classes of infinite permutation groups. Their precision is sharpened by the introduction of a cartesian decomposition concept. This facilitates reduction arguments for primitive groups analogous to those, using orbits and partitions, that reduce problems about general permutation groups to primitive groups. The results are particularly powerful for finite groups, where the finite simple group classification is invoked. Applications are given in algebra and combinatorics to group actions that preserve cartesian product structures. Students and researchers with an interest in mathematical symmetry will find the book enjoyable and useful.

Boolean lattices in finite alternating and symmetric groups

“…We also refer to [30] for a recent thorough investigation on permutation groups admitting Cartesian decompositions, where each of these peculiar examples is thoroughly investigated.…”

A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

“…For the proof of the next lemma we need some basic terminology, which we borrow from [9,Section 4.3 and 4.4]. Let κ be a positive integer and let A be a direct product S 1 × · · · × S κ , where the S i s are pair-wise isomorphic non-abelian simple groups.…”

Transitive characteristically simple subgroups of finite quasiprimitive permutation groups

“…The first part of the following lemma appeared in Scott's paper [Sco80, Lemma p. 328], and is known as Scott's Lemma (see also [PS18b,Section 4.6] for several generalizations). It describes the structure of the subdirect subgroups of a direct product of nonabelian simple groups.…”

“…Finite primitive and quasiprimitive groups were classified by the respective versions of the O'Nan-Scott Theorem; see [PS18b,Chapter 7]. In this classification, we distinguish between 8 classes of finite primitive groups, namely HA, HS, HC, SD, CD, PA, AS, TW, and 8 classes of finite quasiprimitive groups, namely HA, HS, HC, Sd, Cd, Pa, As, Tw.…”

“…Set Γ to be the right coset space [Q 1 : Q 1 ∩ S α ]. By [PS18b,Theorem 4.24], we may assume without loss of generality that G is a subgroup of W = Sym(Γ) wr S r acting in product action on Γ r , and so H can also be viewed as a transitive subgroup of W . By Theorem 1.1, H is a subgroup of the base group Sym(Γ) r of W .…”