External semidirect product

From Groupprops

Definition

Suppose $N$ is a group and $H$ is a group acting on $N$ ; in other words, there is a group homomorphism 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:

$(a, b)(a', b') = (a b . a', b b')$

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.

The external semidirect product becomes an external direct product when the action of $H$ on $N$ is trivial.

External semidirect product

From Groupprops

Definition

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