Let G = {(x; y) : x in R; y > 0}. With (x; y)(u; v) = (x + yu; yv), G is a group  .If topologize G as a subset of 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 be a locally compact group and also subgroup K = {(0; y) : y > 0} be compact?

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?