A survey of symmetric generation of sporadic simple groups

Summary

Abstract

Many of the sporadic simple groups possess highly symmetric generating sets which can often be used to construct the groups, and which carry much information about their subgroup structure. We give a survey of results obtained so far.

Introduction and motivation

This paper is concerned with groups which are generated by highly symmetric subsets of their elements: that is to say by subsets of elements whose set normalizer within the group they generate acts on them by conjugation in a highly symmetric manner. Rather than investigate the behaviour of various known groups, we turn the procedure around and ask what groups can be generated by a set of elements which possesses certain assigned symmetries. It turns out that this approach enables us to define and construct by hand a large number of interesting groups—including many of the sporadic simple groups.

Accordingly we let m*ⁿ denote C_m*C_m* – *C_m, a free product of n copies of the cyclic group of order m. Let F = T₀*T₁* … *T_n−1 be such a group, with T_i = 〈t_i〉 ≅ C_m. Certainly permutations of the set of symmetric generators T = {t₀, t₁, …, t_n−1} induce automorphisms of F. Further automorphisms are given by raising a given t_i to a power of itself coprime to m, while fixing the other symmetric generators. Together these generate the group M of monomial automorphisms of F which is a wreath product H_rS_n, where H_r is an abelian group of order r = Φ(m), the number of positive integers less than m and coprime to it.