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Rigid automorphisms of linking systems

Part of: Representation theory of groups

Published online by Cambridge University Press:  15 March 2021

George Glauberman  and
Justin Lynd Open the ORCID record for Justin Lynd [Opens in a new window]
George Glauberman
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Ave, Chicago, IL60637; E-mail: [email protected]
Justin Lynd
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA70504; E-mail: [email protected]

Abstract

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A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G, then it is of $p'$-order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$-subgroups. We present two applications of this last result, one to tame fusion systems.

MSC classification

Primary: 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure
Secondary: 20D45: Automorphisms
Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e23
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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