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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W. / Wilson, John S.

Managing Editor: Khukhro, Evgenii I. / Kramer, Linus

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Volume 10, Issue 4 (Jul 2007)


On Jordan's theorem for complex linear groups

Michael J Collins
Published Online: 2007-07-26 | DOI: https://doi.org/10.1515/JGT.2007.032


In 1878, Jordan showed that a finite subgroup of GL(n, ℂ) must possess an abelian normal subgroup whose index is bounded by a function of n alone. We will give the optimal bound for all n; for n ⩾ 71, it is (n + 1)!, afforded by the symmetric group S n+1. We prove a ‘replacement theorem’ that enables us to study linear groups by breaking them down into individual primitive constituents and we give detailed information about the structure of the groups that achieve the optimal bounds, for every degree n.

Our proof relies on known lower bounds for the degrees of faithful representations of each quasisimple group, depending on the classification of finite simple groups, through the use of the bounds for primitive groups that the author has previously obtained.

About the article

Received: 2006-09-08

Published Online: 2007-07-26

Published in Print: 2007-07-20

Citation Information: Journal of Group Theory, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/JGT.2007.032.

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