Semidirect products

In general, $${\phi\left(G\right)\cong G/\textrm{Ker}\phi}$$ is not a subgroup of $${G}$$, so we cannot use this equation to decompose $${G}$$ into an internal product by “multiplying both sides by $${\textrm{Ker}\phi}$$.” If $${H}$$ is a subgroup of $${G}$$ and there is a homomorphism $${\phi\colon G\to H}$$ that is the identity on $${H}$$, we can once again extend the arithmetic of groups to state that $${G=\textrm{Ker}\phi\rtimes H}$$, where $${\rtimes}$$ represents the semidirect product (sometimes stated “$${G}$$ splits over $${\textrm{Ker}\phi}$$”).

Note that the semidirect product as defined here is an internal product formed from $${G}$$, and is distinct from the direct product since in general $${(n_{1}h_{1})(n_{2}h_{2})\neq(n_{1}n_{2})(h_{1}h_{2})}$$; in fact non-isomorphic groups can be the semidirect products of the same two constituent groups. If $${H}$$ is normal as well as $${N}$$, then this is not the case and the semidirect product is the same as the internal direct product.

An equivalent definition of the semidirect product starts with a normal subgroup $${N}$$ of $${G}$$, defining $${G=N\rtimes H}$$ if $${G=NH}$$ and $${N\cap H=\mathbf{1}}$$, or equivalently if every element of $${G}$$ can be written in exactly one way as a product of an element of $${N}$$ and an element of $${H}$$. These properties can be seen in a common use of the semidirect product in physics, where $${N}$$ is the group of translations in $${\mathbb{R}^{n}}$$, $${H}$$ is the group of rotations and reflections, and $${N\rtimes H}$$ is thus the group of all rigid transformations.