Semidirect product

The semidirect product is a generalization of the direct product. There are two separate senses in which semidirect product is used, an outer semidirect product and an inner semidirect product.

Outer semidirect product
Given two groups $G$ and $H$ along with a group homomorphism $$\varphi : G \rightarrow \mathrm{Aut}(H)$$ (a homomorphism from $G$ to the group of automorphisms on $H$) the semidirect product $$G\rtimes_{\varphi}H$$ is a group $$(G\times H,\cdot)$$ where

$$(g_0,h_0)\cdot(g_1,h_1) = (g_0\times_G\varphi(h_0)(g_1),h_0\times_H h_1)$$

Inner semidirect product
The inner semidirect product is a special case of the outer semidirect product where $$\varphi = h \mapsto (g \mapsto h\times g\times h^{-1})$$

Examples

 * The direct product is a special case of the outer semidirect product with the homomorphism $$\varphi(h)(g) = g$$.
 * The wreath product is a case of the outer semidirect product. $$G\,{\wr}_n H \cong G^n \rtimes_{\varphi} H$$ where $$\varphi(h) = h^{-1}$$