Centralizer-commutator product decomposition for finite groups and cyclic automorphism group
From Groupprops
Statement
Suppose is a finite group and is a cyclic subgroup of . Let be the subgroup of generated by all elements of the form for . Let denote the subgroup of comprising those elements fixed by every element of . Then:
.
Related facts
- Centralizer-commutator product decomposition for Abelian groups
- Centralizer-commutator product decomposition for finite nilpotent groups
Facts used
- Centralizer of coprime automorphism in homomorphic image equals image of centralizer: Suppose is an automorphism of of order coprime to the order of . Suppose is a normal -invariant subgroup of . Then if denotes the quotient map, we have .
Proof
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Textbook references
- Nilpotent groups and their automorphisms by Evgenii I. Khukhro, ISBN 3110136724, Page 18, Corollary 1.6.4, (Proof uses theorem 1.6.2)^{More info}