Definition 2

Tag 05CT, permanent. Cite it as [Bourbaki, Tag 05CT].

Algebra, chapter I, § 6. Extensions, solvable groups, nilpotent groups

Let $F$ and $G$ be two groups and $\tau$ a homomorphism of $G$ into the automorphism group of $F$ . Write $\tau(g)(f) = gf$ for $g \in G$ and $f \in F$ . The set $F \times G$ with the law of composition

((f, g), (f', g')) \mapsto (f, g) \cdot_{\tau} (f', g') = (f \cdot gf', gg')

is called the external semi-direct product of $G$ by $F$ relative to $\tau$ .

The external semi-direct product of $G$ by $F$ relative to $\tau$ is denoted by $F \times_{\tau} G$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

§ 1. Lie groups

Cites

Nothing.