Large element orders and the characteristic of Lie-type simple groups

Given a matrix group G = 〈X〉 GL(d,q), specified by a set X of generators, it seems that a full structural exploration of G is necessary in order to answer even the simplest questions concerning G , such as finding |G| or testing the membership of any given matrix in G (cf. [LG,BB]). Currently, the standard approach to such an exploration is to set up a recursive scheme of homomorphisms, breaking the input into the image and kernel [LG,NS,O’B,Se]. This reduction bottoms out at an absolutely irreducible matrix group G that is simple modulo scalars. At this terminal stage of the recursion, one finds the name (i.e., the isomorphism type) of G , and then sets up an identification with a standard quasisimple group. For a prime power q = pe , we write ch(q) = p, and for a Lie-type simple group G defined over GF(q) we write ch(G)= p. For a quasisimple matrix group G one proceeds using the following steps.