## Duke Mathematical Journal

### Finite linear groups and bounded generation

#### 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.

#### Article information

Source
Duke Math. J., Volume 107, Number 1 (2001), 159-171.

Dates
First available in Project Euclid: 5 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1091736140

Digital Object Identifier
doi:10.1215/S0012-7094-01-10718-7

Mathematical Reviews number (MathSciNet)
MR1815254

Zentralblatt MATH identifier
1017.20039

Subjects
Primary: 20G40: Linear algebraic groups over finite fields
Secondary: 20E18: Limits, profinite groups

#### Citation

Liebeck, Martin W.; Pyber, L. Finite linear groups and bounded generation. Duke Math. J. 107 (2001), no. 1, 159--171. doi:10.1215/S0012-7094-01-10718-7. https://projecteuclid.org/euclid.dmj/1091736140

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