Computing with simple groups: maximal subgroups and presentations

Abstract

For the non-abelian simple groups G of order up to 10⁶ , excluding the groups PSL(2,q), q > 9, the presentations in terms of an involution a and an element b of minimal order (with respect to a) such that G=<a,b> are well known. The presentations are complete in the sense that any pair (x,y) of generators of G satisfying x²=yᵐ=1, with m minimal, will satisfy the defining relations of just one presentation in the list. There are 106 such presentations. Using a computer, we give generators for each maximal subgroup of the groups G. For each presentation of G, the generators of maximal subgroups are given as words in the group generators. Similarly generators for a Sylow p-subgroup of G, for each p, are given. For each group G, we give a representative for each conjugacy class of the group as a word in the group generators. Minimal presentations for each Sylow p-subgroup of the groups G, and for most of the maximal subgroups of G are constructed. To obtain such presentations, the Schur multipliers of the underlying groups are calculated. The same tasks are carried out for those groups PSL(2,q) of order less than 10⁶ which are included in the "ATLAS of finite groups". For these groups we consider a presentation on two generators x, y with x²=y³=1. A finite group G is said to be efficient if it has a presentation on d generators and d+rank(M(G)) relations (for some d) where M(G) is the Schur multiplier of G. We show that the simple groups J₁, PSU(3,5) and M₂₂ are efficient. We also give efficient presentations for the direct products A₅xA₆, A₅xA₆,A₆xA₇ where Ĥ denotes the covering group of H.