External semidirect product
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.
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 for groups
- External direct product (corresponding internal notion: internal direct product, equivalence)
- External wreath product (corresponding internal notion: internal wreath product)