# Centralizer-commutator product decomposition for finite groups and cyclic automorphism group

## Statement

Suppose $G$ is a finite group and $H$ is a cyclic subgroup of $\operatorname{Aut}(G)$. Let $[G,H]$ be the subgroup of $G$ generated by all elements of the form $g\sigma(g)^{-1}$ for $\sigma \in H$. Let $C_G(H)$ denote the subgroup of $G$ comprising those elements fixed by every element of $H$. Then:

$G = [G,H]C_G(H)$.

## Facts used

1. Centralizer of coprime automorphism in homomorphic image equals image of centralizer: Suppose $\varphi$ is an automorphism of $G$ of order coprime to the order of $G$. Suppose $N$ is a normal $\varphi$-invariant subgroup of $G$. Then if $\pi:G \to G/N$ denotes the quotient map, we have $\pi(C_G(\varphi)) = C_{G/N}(\varphi)$.