Internal semidirect product

This article describes a product notion for groups. See other related product notions for groups.

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
VIEW: Definitions built on this | Facts about this: (facts closely related to Internal semidirect product, all facts related to Internal semidirect product) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a complete list of semi-basic definitions on this wiki

Definition

Definition with symbols

A group $G$ is termed an internal semidirect product of subgroups $N$ and $H$ if the following hold:

$N$ is a normal subgroup of $G$
$N$ and $H$ are permutable complements

Note here that $H$ acts as automorphisms on $N$ by the conjugation action.

If we start out with $N$ and $H$ as abstract groups, and with the action of $H$ on $N$ (abstractly) which comes from the conjugation in $G$ , then the external semidirect product formed from these is isomorphic to $G$ .

Terminology

A subgroup which occurs as the normal subgroup for an internal semidirect product is termed a split normal subgroup
A subgroup which occurs as the permutable complement to a normal subgroup, is termed a retract. This is because there is a retraction from the whole group, to this subgroup, whose kernel is the normal subgroup.

Relation with other properties

Stronger product notions

Weaker product notions