Introduction

It is well-known that the modular group is isomorphic to the free product of the groups of order 2 and 3, and therefore a group appears as factor-group of if and only if it is generated by an involution and an element of order 3. Such groups are often called -generated. The following finite groups are known to be -generated:

• the alternating groups , (Miller 1901, see references in [DMT]);
• the projective groups for ([Mac]); for ([Gar] and [Co]); the linear groups for odd ([TV]), for , odd and ([DMV1] and [DMV2]); for ([Tamb], see also [DMV1] and [DMV2]);
• the Chevalley groups and the twisted groups ([Mal1],[Mal2]);
• all sporadic simple groups except , , and ([Wo]).

The above mentioned results together with evidence provided by work currently under way, support the conjecture that most finite simple groups appear as factor-groups of .

Some of the exceptions listed above are true exceptions, like, say, for , or the sporadics; whereas most of the other exceptions are only due to the methods used in the proofs. Also, these proofs suggest that, while a uniform treatment should be at hand for all classical groups when the Lie rank is large enough, the small dimensional cases require a somewhat different choice for the generators, and therefore a special ``ad hoc" analysis.

The present paper, as part of work presently under way, covers the ``smallest" symplectic groups for , . A pair of generators is exhibited, which appears to be very apt to give the desired result via the knowledge of maximal subgroups. On the other hand, this pair does not easily provide a transvection in , which would have allowed to try, alternatively, to make use of the classification of groups generated by transvections (as in [TV] and [DMV1],[DMV2]).

As a final remark, we observe that a -generated simple group can be generated by three conjugate involutions. (It is known that every finite non-abelian simple group, apart from , can be generated by three involutions, see the recent paper [MSW], and the papers quoted there.)