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Groupprops β

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.