Article contents
Rigid automorphisms of linking systems
Published online by Cambridge University Press: 15 March 2021
Abstract
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
- Type
- Algebra
- Information
- Creative Commons
- 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
References
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