Finite linear groups and bounded generation

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