# External semidirect product

From Groupprops

## Contents

## Definition

### Definition with the left action convention

Suppose is a group and is a group acting on ; in other words, there is a group homomorphism , from to the automorphism group of . The **external semidirect product** of and , denoted is, as a set, the Cartesian product , with multiplication given by the rule:

Writing the action , we get:

The way multiplication is defined, it turns out that:

- embeds as a normal subgroup of (via ) and embeds as a subgroup via . The two subgroups are permutable complements, hence the external semidirect product is the same as an internal semidirect product once we identify and with their images in . In particular, the image of is a complemented normal subgroup in and the image of is a retract of .
- The action of the image of , on the image of , via
*conjugation*in , is the same as the abstract action that we started with.

## Comments

### Case of abelian normal subgroup

In the special case where is an abelian group and the binary operation of is denoted additively, the multiplication rule for can be written as:

This notation comes up in the study of the second cohomology group.

### Case of trivial action

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

## Related notions

### Related notions for groups

- External direct product (corresponding internal notion: internal direct product, equivalence)
- External wreath product (corresponding internal notion: internal wreath product)