# External semidirect product of semigroup and group

## Definition

### Definition with the left action convention

Suppose is a semigroup and is a group along with a homomorphism (the automorphism group of ). The **external semidirect product** of by with respect to , denoted , is defined as the following semigroup . As a set , with the multiplication given by:

If we denote by , this can be rewritten as: