Proposition 1

The left and right translations are isomorphisms of the right uniformity onto itself.

As to the right translations, the result is clear, since the relation $yx^{-1} \in V$ is equivalent to $(ya)(xa)^{-1} \in V$ [in other words, the mapping $(x, y) \to (xa, ya)$ leaves $V_d$ fixed]. For the left translations, the result follows from $(GV_{\text{III}})$; for $yx^{-1} \in V$ if and only if $(ay)(ax)^{-1} \in aV a^{-1}$; hence $x \to ax$ is uniformly continuous on $G_d$.

Similarly, the right and left translations are isomorphisms of the left uniformity onto itself.

Every inner automorphism $x \to axa^{-1}$ of $G$ is therefore an automorphism for the group structure of $G$, for the topology of $G$, and for both the uniformities of $G$.