Notation and statement of results

Let , a prime, . Let be the 4-dimensional vector space over the finite field . Let be a non-degenerate symplectic form on . As usual, we denote by the group of all isometries of the symplectic space , and by its projective image, i.e. the group acting faithfully on the projective space associated to . Moreover, we will denote by the ``general" symplectic group, i.e. the group of all similarities of , and by its projective image. (The notation for the other classical groups appearing in the sequel is also standard. The reader can refer to [Asch1] or [KL].)

By choosing a hyperbolic basis in , we may assume that is represented by the matrix and we may identify with the group of all matrices such that .

Now let and , where . Then , , , and .

It is our aim to prove that is generated by and , provided that . As an immediate consequence of this fact we obtain the following

that, however, is by no means -generated (cf. [DMT]).

The next paragraph, though long, is necessary in order to have a picture of the subgroup structure of in such terms that are most convenient for our purposes in 3.