Finite Linear Groups and Bounded Generation

We extend a result of E. Hrushovski and A. Pillay as follows. Let G be a finite subgroup of GL(n,F) where F is a field of characteristic p such that p is sufficiently large compared to n. Assume thatG is generated by p-elements. ThenG is a product of 25 of its Sylow p-subgroups. If G is a simple group of Lie type in characteristic p, the analogous result holds without any restriction on the Lie rank of G. We also give an application of the Hrushovski-Pillay result showing that finitely generated adelic profinite groups are boundedly generated (i.e., such a group is a product of finitely many closed procyclic subgroups). This confirms a conjecture of V. Platonov and B. Sury which was motivated by characterizations of the congruence subgroup property for arithmetic groups.