# Definition:Semidirect Product

## Definition

Let $H$ and $N$ be groups.

Let $\Aut N$ denote the automorphism group of $N$.

Let $\phi: H \to \Aut N$ be a group homomorphism, that is, let $H$ act on $N$.

The (outer) semidirect product of $N$ and $H$ with respect to $\phi$ is the cartesian product $N \times H$ with the group law defined as:

$\tuple {n_1, h_1} \circ \tuple {n_2, h_2} = \tuple {n_1 \cdot \map \phi {h_1} \paren {n_2}, h_1 \cdot h_2}$

It is denoted $N \rtimes_\phi H$.