Finite subgroups of algebraic groups

Larsen, Michael; Pink, Richard

doi:10.1090/S0894-0347-2011-00695-4

Finite subgroups of algebraic groups
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by Michael J. Larsen and Richard Pink;

J. Amer. Math. Soc. 24 (2011), 1105-1158

DOI: https://doi.org/10.1090/S0894-0347-2011-00695-4

Published electronically: April 28, 2011

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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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