\begin{definition}[Definition:Semidirect Product]
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 operation defined as:
:$\tuple {n_1, h_1} \circ \tuple {n_2, h_2} = \tuple {n_1 \cdot \map \phi {h_1} \paren {n_2}, h_1 \cdot h_2}$
{{explain|What specifically is "$\cdot$" here? This is an instance where it is important (in my belief) to be completely explicit as to what the operations are within the groups $H$ and $N$. It may also be clearer to refer to $G_1$ and $G_2$ because $N$ and $H$ have connotations of "normal subgroup" and "subgroup" respectively, and it is highly desirable to be consistent with other pages.}}
It is denoted $N \rtimes_\phi H$.
\end{definition}