Proposition 28

*Let L, N be two topological groups, let $y \to \sigma_y$ be a homomorphism of L into the group of automorphisms $\Gamma$ of the (non-topological) group structure of N; and suppose that the mapping $(x, y) \to \sigma_y(x)$ of $N \times L$ into N is continuous. Then:

1) On the external semi-direct product S of N and L, relative to $\sigma$, the product of the topologies of N and L is compatible with the group structure; the canonical injections $j_1 : N \to S$ and $j_2 : L \to S$ are isomorphisms of the topological groups N and L respectively onto the subgroups $j_1(N)$ and $j_2(L)$ of S, and $pr_2$ is a strict morphism of S onto L.

2) Let $f : N \to G$ and $g : L \to G$ be two continuous homomorphisms into a topological group G, satisfying (4); then the homomorphism

of S into G is continuous.

This is an immediate consequence of the definitions and of the properties of the product topology.

The topological group S thus defined is said to be the external topological semi-direct product of N and L (relative to σ); notice that the condition imposed on σ implies that L operates continuously on the left on N according to the external law $(x, y) \to \sigma_y(x)$ [no. 4].

Now let G be a topological group and let N and L be two subgroups of G such that G is the semi-direct product of N and L, qua non-topological group; it is then clear that the mapping $(x, y) \to \sigma_y(x)$ is continuous on $N \times L$, and that the canonical bijective homomorphism

of S onto G is continuous. But this homomorphism is not necessarily bicontinuous; when it is bicontinuous, G is said to be the topological semi-direct product of N and L. For this to be so it is necessary and sufficient that, if $p : G \to N$ and $q : G \to L$ are the mappings which make correspond to $z \in G$ the unique elements $p(z) \in N$ and $q(z) \in L$ such that $z = p(z)q(z)$, then one of the mappings $p, q$ is continuous (in which case both are continuous). It comes to the same thing to say that the restriction to L of the canonical mapping $G \to G/N$ is an isomorphism of the topological group L onto the topological group G/N.