Let $G = \{(x; y) : x \in \mathbb{R}, y > 0\}$. With $(x, y)(u, v) = (x + yu, yv)$, $G$ is a group. If we topologize $G$ as a subset of $\mathbb{R}^2$, it is known that $G$ is a locally compact group that is not unimodular (see (15.17) of Hewitt-Ross). Is there another topological structure of the group $G$ such that $G$ is a locally compact group and the subgroup $K = \{(0, y) : y > 0\}$ is compact?