Semi-direct product

The (external) semi-direct product, in group theory, is a generalization of the direct product.

Two Equivalent Definitions

Let $E$ be a group, $F$ a normal subgroup of $E$ , and $G$ a subgroup of $E$ . If $E = FG$ and $F \cap G = \{e\}$ , then $E$ is called the (left) (external) semi-direct product of $F$ and $G$ .

Since $F$ is normal, the restriction of each inner automorphism of $E$ to $F$ is an automorphism of $F$ . In particular, there exists a function $\tau$ which associates each element of $G$ with an automorphism on $F$ (namely, the restriction to $F$ of the inner automorphism on $E$ ). Then $E$ is called the (external) semi-direct product of $G$ by $F$ relative to $\tau$ and is denoted $F \times_\tau G$ . Each element of $E$ is identified with its corresponding element of $F \times G$ , and the group law on $E$ is defined as $(f,g)(f',g') = (f \cdot {^{g}f'}, gg'),$ for $fgf'g' = f(gfg^{-1}) \cdot gg' = f \cdot {^g f'} \cdot gg' .$

Conversely, let $F$ and $G$ be groups, and let $\tau$ be a homomorphism from $G$ into the group of automorphisms of $F$ . The set $F\times G$ under the operation $(f,g)(f',g') = (f \cdot {^{g}f'}, gg')$ is a group; it is $F \times_\tau G$ . Indeed, $\begin{align*} \bigl( (f,g)(f',g') \bigr) (f'',g'') &= (f \cdot {^g f'},gg')(f'',g'') = (f \cdot {^g f'} \cdot {^{gg'} f''}, gg'g'') \\ &= (f,g)(f' \cdot {^{g'} f''}, g'g'') = (f,g) \bigl( (f',g') (f'',g'') \bigr), \end{align*}$ so the law of composition is associative; the identity is $(e,e)$ ; and the inverse of $(f,g)$ is $( {^{g^{-1}}f^{-1}}, g^{-1} )$ .

Semi-direct products and extensions

Evidently, if $E$ is a semidirect product of $G$ by $F$ , then it is a group extension of $G$ by $F$ with a section (the projection onto $G$ ). The converse is also true. Indeed, if $\mathcal{E} : F \stackrel{i}{\to} E \stackrel{p}{\to} G$ be an extension of $G$ by $F$ with a section $s: G \to E$ , then $E = i(F)s(G)$ , and $i(F) = \text{Ker}(i)$ is a normal subgroup of $E$ .

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Semi-direct product

Two Equivalent Definitions

Semi-direct products and extensions

See also