Probabilistic methods in the generation of finite simple groups

Summary

Abstract

We survey recent progress, made using probabilistic methods, on several problems concerning generation of finite simple groups. For example, we outline a proof that all but finitely many classical groups different from PSp₄{q) (q = 2^a or 3^a) can be generated by an involution and an element of order 3.

Results

In this survey we present some new methods and results in the study of generating sets for the finite (nonabelian) simple groups. The results are largely taken from the three papers [16], [17], [18]. We shall present the results in this first section, and outline some proofs in sections 2 and 3. We begin by describing some of the basic questions and work in the area.

It is a well known consequence of the classification that every finite simple group can be generated by two elements. This result was established early this century for the alternating groups by Miller [23] and for the groups PSL₂(q) by Dickson [8]. Various other simple groups were handled by Brahana [5] and by Albert and Thompson [1], but it was not until 1962 that Steinberg [26] showed that all finite simple groups of Lie type can be generated by two elements. To complete the picture, in 1984 Aschbacher and Guralnick [2] established the same conclusion for sporadic groups.

A refinement of the two element generation question asks whether every finite simple group can be generated by an involution and a further element. Partial results on this question were obtained in the above-mentioned papers [23], [5], [1], [2], but only recently has the question been answered completely, in the affirmative, by Malle, Saxl and Weigel [22].