Rigid automorphisms of linking systems
Forum of Mathematics, Sigma, Tome 9 (2021)
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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.
@article{10_1017_fms_2021_17,
author = {George Glauberman and Justin Lynd},
title = {Rigid automorphisms of linking systems},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {9},
year = {2021},
doi = {10.1017/fms.2021.17},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2021.17/}
}
George Glauberman; Justin Lynd. Rigid automorphisms of linking systems. Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2021.17
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