uniform structure of a topological group

Let $G$ be a topological group. There is a natural uniform structure on $G$ which induces its topology. We define a subset $V$ of the Cartesian product $G\times G$ to be an entourage if and only if it contains a subset of the form

$$V_N=\{(x,y)\in G\times G:x{y}^{-1}\in N\}$$

for some $N$ neighborhood of the identity element. This is called the right uniformity of the topological group, with which multiplication becomes a uniformly continuous map.The left uniformity is defined in a fashion, but in general they don’t coincide, although they both induce the same topology on $G$.