Computing with matrix groups∗

A group is usually input into a computer by specifying the group either using a presentation or using a generating set of permutations or matrices. Here we will emphasize the latter approach, referring to [Si3, Si4, Ser1] for details of the other situations. Thus, the basic computational setting discussed here is as follows: a group is given, specified as G = 〈X〉 in terms of some generating set X of its elements, where X is an arbitrary subset of either Sn or GL(d, q) (a familiar example is the group of Rubik’s cube). The goal is then to find properties of G efficiently, such as |G|, the derived series, a composition series, Sylow subgroups, and so on. When G is a group of permutations there is a very well-developed body of literature and algorithms for studying its properties (see Section 2). The matrix group situation is much more difficult, and is the focus of the remaining sections of this brief survey. Sections 4 and 5 discuss the case of simple groups, and section 6 uses these to deal with general matrix groups. We will generally emphasize the group-theoretic aspects of the subject, rather than ones involving implementation in the computer systems GAP [GAP4] or Magma [BCP]. Thus, the word “efficiently” used above will usually mean for us “in time polynomial in the input length of the problem” rather than “works well in practice”. One can ask for the relevance of such questions to finite group theory. Certainly computers have been involved in the construction of sporadic simple groups, as well as in the study of these and other simple groups. We will make a few comments concerning the expected uses in GAP and Magma of the results presented here. However, our point of view includes a slightly different aspect: the purely mathematical questions raised by computational needs have led to new points of view and new questions concerning familiar groups.