# External semidirect product of semigroup and group

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Suppose $S$ is a semigroup and $H$ is a group along with a homomorphism $\rho:H \to \operatorname{Aut}(S)$ (the automorphism group of $S$). The external semidirect product of $S$ by $H$ with respect to $\rho$, denoted $S \rtimes H$, is defined as the following semigroup $T$. As a set $T = S \times H$, with the multiplication given by:
$\! (a,b)(a',b') = (a (\rho(b)(a')), bb')$
If we denote $\rho(b)(a')$ by $b \cdot a'$, this can be rewritten as:
$\! (a,b)(a',b') = (a (b \cdot a'), bb')$