External semidirect product

From Groupprops

Definition

Definition with the left action convention

Suppose $N$ is a group and $H$ is a group acting on $N$ ; in other words, there is a group homomorphism $\rho:H \to \operatorname{Aut}(N)$ , from $H$ to the automorphism group of $N$ . The external semidirect product $G$ of $N$ and $H$ , denoted $N \rtimes H$ is, as a set, the Cartesian product $N \times H$ , with multiplication given by the rule:

$\! (a,b)(a',b') = (a(\rho(b)(a')),bb')$

Writing the action $\rho(b)a' = b \cdot a'$ , we get:

$(a,b)(a',b') = (a(b \cdot a'),bb')$

The way multiplication is defined, it turns out that:

$N$ embeds as a normal subgroup of $G$ (via $a \mapsto (a,e)$ ) and $H$ embeds as a subgroup via $b \mapsto (e,b)$ . The two subgroups are permutable complements, hence the external semidirect product is the same as an internal semidirect product once we identify $N$ and $H$ with their images in $G$ .
The action of the image of $H$ , on the image of $N$ , via conjugation in $G$ , is the same as the abstract action that we started with.