Does there exist a locally compact, Hausdorff, non-discrete topological group $G$ with the following properties?
- There exists $H \leq G$ proper clopen subgroup;
- $G / H$ is an uncountable set.
2026-03-29 17:25:50.1774805150
Subgroup of a locally compact group with uncountably many laterals.
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Take $K$ any uncountable discrete group (e.g. $\mathbb{R}$ with discrete topology) and $H$ any locally compact Hausdorff non-discrete group (e.g. $\mathbb{R}$ with Euclidean topology). Then $G=K\times H$ seems to satisfy your conditions.