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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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
MR Author ID:
293592
Email:
mjlarsen@indiana.edu
Richard Pink
Affiliation:
Department of Mathematics, ETH Zürich, CH - 8092 Zürich, Switzerland
MR Author ID:
139765
Email:
pink@math.ethz.ch
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