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Finite subgroups of algebraic groups


Authors: Michael J. Larsen and Richard Pink
Journal: J. Amer. Math. Soc. 24 (2011), 1105-1158
MSC (2010): Primary 20G40
DOI: https://doi.org/10.1090/S0894-0347-2011-00695-4
Published electronically: April 28, 2011
MathSciNet review: 2813339
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Abstract: Generalizing a classical theorem of Jordan to arbitrary characteristic, we prove that every finite subgroup of $ \operatorname{GL}_n$ over a field of any characteristic $ p$ possesses a subgroup of bounded index which is composed of finite simple groups of Lie type in characteristic $ p$, a commutative group of order prime to $ p$, and a $ p$-group. While this statement can be deduced from the classification of finite simple groups, our proof is self-contained and uses methods only from algebraic geometry and the theory of linear algebraic groups. We believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics.


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Additional Information

Michael J. Larsen
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: mjlarsen@indiana.edu

Richard Pink
Affiliation: Department of Mathematics, ETH Zürich, CH - 8092 Zürich, Switzerland
Email: pink@math.ethz.ch

DOI: https://doi.org/10.1090/S0894-0347-2011-00695-4
Received by editor(s): August 20, 2010
Received by editor(s) in revised form: September 2, 2010, and January 27, 2011
Published electronically: April 28, 2011
Additional Notes: The first author was partially supported by a Sloan grant and by NSF grants DMS-9727553 and DMS-0800705.
Article copyright: © Copyright 2011 American Mathematical Society

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