JOURNAL OF ALGEBRA 116, 316-333 (1988) On the Generation of Finite Simple Groups by Pairs of Subgroups M . CHIARATAMBURINI* Dipartimento di Matematica, Universitci di Milano, Via C. Saldini 50, Milan, Italy AND JOHNS. WILSON Christ’s College, Cambridge, England Communicated by Peter M. Neumann Received November 11, 1986 1. INTRODUCTION Let A and B be finite groups which are non-trivial and not both of order 2. It seems likely that every finite simple group S of sufficiently large rank can be generated by a subgroup isomorphic to A and a subgroup isomorphic to B. 0,ur object here is to show that, provided that 1 A ( and ) BI are not too small, every projective special linear group and alternating group of sufficiently large rank can be generated by a copy of A and a copy of B. More precisely, we shall prove the following two results: THEOREM 1. If 1 A I( BI > 12 und y is uny prime power, then for ull n 2 4 1 A 11 B 1 + 12, the group PSL,(q) has subgroups 2 E A and BE B such - - that PSL,(q) = (A, B). THEOREM 2. If 1 A I I B I > 12, then for all n > 4 1 A I I B I + 12 the alter- - - nating group A,, has subgroups d z A and i? z B such that A, = (A, B). The case in which A and B are cyclic is of course of particular interest, since it yields pairs of generating elements of prescribed orders. The fact that every group PSL,(q) can be generated by two elements, one of which is an involution, was proved by Albert and Thompson [ 11. As an * Work done as part of the C.N.R. research programme with financial support from the M.P.I. 316 0021~8693/88 $3.00 Copyright 0 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.