Proposition 20

Let $A$ be a subgroup of a topological group $G$, let $H$ be a normal subgroup of $G$, and let $\varphi$ denote the canonical mapping of $G$ onto $G/H$. Then the canonical bijection of $\varphi(A)$ onto $AH/H$ is an isomorphism of topological groups.

This follows from the preceding remarks, and from Proposition 10 of no. 4.

The canonical bijection of $A/(A \cap H)$ onto $\varphi(A)$ is a continuous homomorphism, since it arises from the restriction of $\varphi$ to $A$ by passing to the quotients; but in general the topological groups $A/(A \cap H)$ and $AH/H$ are not isomorphic (cf. § 4, no. 1, Proposition 1, Corollary 3).

\* For example, take $G$ to be the additive group $\mathbf{R}$ of real numbers, take $H$ to be the group $\mathbf{Z}$ of integers, and take $A$ to be the group $\theta \mathbf{Z}$ of integer multiples of an irrational number $\theta$. Then $A \cap H = \{0\}$, so that $A/(A \cap H)$ is a discrete group, isomorphic to $\mathbf{Z}$; on the other hand, $A + H$ is dense in $\mathbf{R}$ (as we shall see in Chapter V, § 1, no. 1, Proposition 1), hence $(A + H)/H$, which is locally isomorphic to $A + H$ (no. 6, Proposition 19), is not a discrete group and consequently is not isomorphic to $A/(A \cap H)$. \*

Nevertheless, we have the following proposition: