semidirect product

Definition

Let be a group that acts on a group via the homomorphism

Then the semidirect product is the group with the following properties:

1. Elements are from the set .
2. Multiplication is “twisted by ” by the equation

Relation to direct product

If the action of on is trivial, then the semidirect product gives multiplication

Which is equivalent to the direct product.

Interior semidirect products

There are equivalent notions of exterior (defined above) and interior semidirect products. For a group , let be a normal subgroup and be a subgroup such that every element of can be written uniquely in the form for and then

The equivalence means that for any semidirect product then is a normal subgroup and is a subgroup satisfying the condition above.

Group extensions/ Splitting SES

Semidirect products of groups give a way to split short exact sequences of groups. Consider the short exact sequence

We say that is an extension of by .

The sequence splits if there exists a morphism such that . The diagram for the situation is

Split extensions of of by are semidirect products of groups for some action .