Results on two-element generation have been established for many important families of finite groups, encompassing all finite simple groups. For example, theorems of this type were proved for the finite non-abelian simple groups of Lie type, the symplectic groups over finite fields, and the sporadic simple groups by Steinberg [13], Stanek [12], and Aschbacher and Guralnick [2], respectively. A comprehensive survey of the status of results on this and related generation problems, some historical remarks, and an extensive list of references to the original literature can be found in a recent paper by DiMartino and Tamburini [6]. The purpose of the present paper is to report on some recent results on the generation of the finite orthogonal groups, including a proof that all such groups are generated by two elements. This proof is contained in joint work of Ishibashi and the author [10], except for several low-dimensional cases which are resolved in [4].

Background

The development of a general theory of the classical groups over arbitrary fields and division rings came to fruition in Dieudonné's fundamental volume La Géométrie des Groupes Classiques [5]. The definitive book of Hahn and O'Meara [7] provides a thorough modern treatment of this theory in the broader context in which the underlying ring of scalars is kept as general as possible.