Exercise 3, § 3

Let $G$ be a (not necessarily commutative) linearly ordered group. If $G$, endowed with the topology $\mathscr{T}_0(G)$, is locally compact and not discrete, then $G$ is locally isomorphic to $\mathbf{R}$, and the identity component of $G$ is an open subgroup isomorphic to $\mathbf{R}$ (use Exercise 6 of Chapter IV, § 2, and Proposition 2 of § 2).