Proposition 3

The external semi-direct product $F \times_{\tau} G$ is a group. The mappings $i: F \to F \times_{\tau} G$ defined by $i(f) = (f, e)$, $p: F \times_{\tau} G \to G$ defined by $p(f, g) = g$, and $s: G \to F \times_{\tau} G$ defined by $s(g) = (e, g)$ are group homomorphisms. The triple $(F \times_{\tau} G, i, p)$ is an extension of $G$ by $F$ and $s$ is a section of the extension.

We have:

Now $g(f'.g'f'') = gf'.gg'f''$, which shows that the law of composition defined by (1) is associative. The element $(e, e)$ is the identity under this law. The element $(f, g)$ admits as inverse $(g^{-1}f^{-1}, g^{-1})$. Hence the law of composition on $F \times_{\tau} G$ is a group law. The other assertions are immediate.

Using the notation of Proposition 3, $\mathscr{E}_{\tau}$ will denote the extension

Let $\mathscr{E}' : F \xrightarrow{i'} E' \xrightarrow{p'} G$ be an extension of $G$ by $F$ and $s' : G \to E'$ a section of $\mathscr{E}'$. We define an operation $\tau$ of $G$ on the group $F$ by:

(2)