Exercise 13, § 1

Let $G$ be a locally compact, non-compact topological group. Show that the uniformity induced on the product space $H = G \times G$ by the uniformity of its Stone-Čech compactification $\beta H$ is strictly finer than the uniformity induced by that of $(\beta G) \times (\beta G)$. [Assuming the result false, show, by applying Exercise 7 a) to the mapping $(x, y) \to xy^{-1}$ of $H$ into $G$, that $G$ would be isomorphic to a subgroup of a compact group; now apply Chapter III, § 3, no. 3, Proposition 4, Corollary 1.]