Minimal parabolic systems for the symmetric and alternating groups

Summary

Abstract

We determine the minimal parabolic subgroups and minimal parabolic systems of the alternating and symmetric groups with respect to a Sylow 2-subgroup.

Introduction

Using the finite groups of Lie type as a role model Ronan and Smith [5] and Ronan and Stroth [6] studied certain subgroups of the sporadic finite simple groups. The underlying idea was to obtain geometric structures analogous to buildings via suitable systems of subgroups in each of these groups. Here we investigate the alternating and symmetric groups for the prime 2 from a perspective similar to [6], the emphasis being on minimal parabolic subgroups, which we now define.

Let G be a finite group, p a prime dividing the order of G and T ∈ Syl_p(G). Put B = N_G(T). Then a subgroup P of G which properly contains B is called a minimal parabolic subgroup (of G) with respect to B if B is contained in a unique maximal subgroup of P. Note that, unlike Ronan and Stroth [6], here we do not require that O_P(P) ≠ 1. We denote the collection of minimal parabolic subgroups of G with respect to B by M(G, B). In the case when G is a finite group of Lie type defined over a field of characteristic p, the subgroups in M(G, B) are indeed minimal parabolic subgroups of G and our nomenclature coincides with the standard one.

Let P₁, …, P_m ∈ M(G, B). Then we call S = {P₁, …, P_m} a minimal parabolic system of rank m provided that G = 〈S〉 and no proper subset of S generates G.