# Commutator of element and automorphism

## Definition

### Basic definition

Let be a group, be an element, and be an automorphism of . Then, the commutator of and , denoted is sometimes defined as (for those following the left-action convention):

Under the right-action convention, the commutator is written as and is defined as:

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

### Definition as an ordinary commutator of elements

Let be a group, be an element and . Consider the external semidirect product (we can take the semidirect product of with any subgroup of containing ). The commutator of and is simply the commutator of these as elements in the semidirect product.