Commutator of element and automorphism

Definition

Basic definition

Let $G$ be a group, $a\in G$ be an element, and $\sigma \in \operatorname {Aut} (G)$ be an automorphism of $G$. Then, the commutator of $a$ and $\sigma$, denoted $[a,\sigma ]$ is sometimes defined as (for those following the left-action convention):

$a\sigma (a)^{-1}$

Under the right-action convention, the commutator is written as $[\sigma ,a]$ and is defined as:

$\sigma (a)^{-1}a$

The notion of commutator gives the usual notion of commutator of two elements $a$ and $b$, if we take $\sigma$ as conjugation by $b$ (left and right notions respectively).

Definition as an ordinary commutator of elements

Let $G$ be a group, $a\in G$ be an element and $\sigma \in \operatorname {Aut} (G)$. Consider the external semidirect product $G\rtimes \langle \sigma \rangle$ (we can take the semidirect product of $G$ with any subgroup of $\operatorname {Aut} (G)$ containing $\sigma$). The commutator of $a$ and $\sigma$ is simply the commutator of these as elements in the semidirect product.