Uniform structure in topological groups

Abstract:

A characterization is given of those locally compact groups having equivalent right and left uniform structures. It is shown that $\alpha$-compact locally compact groups have equivalent uniformities iff for each right uniformly discrete set $B$ such that card $(B) \leq \alpha$ and each neighborhood $U$ of $e,{ \cap _{x \in B}}xU{x^{ - 1}}$ is a neighborhood of $e$. It is also shown that a group is not unimodular iff it contains an open $\sigma$-compact group which is not unimodular.