Abstract
We extend a result of E. Hrushovski and A. Pillay as follows. Let G be a finite subgroup of GL(n,$\mathbb{F}$) where $\mathbb{F}$ is a field of characteristic p such that p is sufficiently large compared to n. Assume that G is generated by p-elements. 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 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.
Citation
Martin W. Liebeck. L. Pyber. "Finite linear groups and bounded generation." Duke Math. J. 107 (1) 159 - 171, 1 March 2001. https://doi.org/10.1215/S0012-7094-01-10718-7
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