Centralizer of coprime automorphism in homomorphic image equals image of centralizer
From Groupprops
This article states and (possibly) proves a fact about a finite group and a Coprime automorphism group (?): a subgroup of the automorphism group whose order is relatively prime to the order of the group itself.
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Statement
Suppose is a finite group and is an automorphism of whose order is coprime to the order of . Suppose is a normal -invariant subgroup of , and let denote the quotient map. Then, has a natural induced action on the quotient group , and we have:
.
Related facts
Applications
- Stability group of subnormal series of finite group has no other prime factors
- Centralizer-commutator product decomposition for finite groups
- Burnside's theorem on coprime automorphisms and Frattini subgroup
References
Textbook references
- Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, Page 17, Theorem 1.6.2, ^{More info}