The factorizations of the finite exceptional groups of lie type

Abstract

A finite group is said to be factorizable if G = AB for some proper subgroups A and B of G; such an expression is called a factorization of G. Factorizations have been the subject of considerable study-see, for example, [ 11, 121. For the case where G is a finite simple group some results can be found in [8, 20, 26, 27, 291. In this paper we determine completely the factorizations of the exceptional simple groups of Lie type and their automorphism groups. (Here, by an exceptional simple group of Lie type we mean a non-abelian finite simple group associated with one of the families G,, F4, E,, E,, E,, ‘B,, 2G2, 3D,, ‘F4, and 2E,, excluding G,(2)’ and *G*(3) in view of the isomorphisms G,(2)’ E U,(3) and ‘G,(3) Z L,(8).) There are two interesting features of our main result. One is that remarkably few of the exceptional simple groups of Lie type are factorizable-for example, E,(q), *EC(q), E,(q), and E,(q) possess no factorizations. In contrast, some striking factorizations are found to exist for the groups G,(q) and F,(q). (For G,(q) these factorizations are already known [20, 271.) In another paper [ 151 the maximal factorizations of all the other finite simple groups and their automorphism groups are classified (where the factorization G = AB is maximal if both A and B are maximal subgroups of G). The results of this paper and [ 151 are used in [ 161 to obtain a classification of the maximal subgroups of the finite alternating and symmetric groups. Our main results here are the following two theorems.