Let G be any of the groups (P )GL(n, q), (P ) SL(n, q). Define a (simple) graph Γ = Γ (G) on the set of elements of G by connecting two vertices by an edge if and only if they generate G. Suppose that n is at least 12. Then the maximum size of a complete subgraph in Γ is equal to the chromatic number of Γ if n ≡ 2 (mod 4), or if n ≡ 2 (mod 4), q is odd and G = (P ) SL(n, q). This work was motivated by a question of Blackburn.

A construction of the Monster simple group is described implicitly as 196882 × 196882 matrices over the field of 3 elements.

We use our computer construction of the Monster sporadic simple group to find a new maximal subgroup PGL 2 (29). In particular, we prove containment of L 2 (29)

A set of proper subgroups is a covering for a group if their union is the whole group. Determining the size of a smallest covering is an open problem for many simple groups. For some of the sporadic groups, we find subgroup coverings of minimal cardinality. For others we specify the isomorphism types of subgroups in a smallest covering and use graphs to calculate bounds for its size.

We present an algorithm to reduce the constructive membership problem for a black-box group G to three instances of the same problem for involution centralisers in G. If G is a simple group of Lie type in odd characteristic, then this reduction can be performed in (Monte Carlo) polynomial time.