the Sorgenfrey line $ mathbb S $ is the actual line with the topology generated by the base composed of all half-intervals $[Ab)$[Ab)$[ab)$[ab)$ for real numbers $ a <b $.
The Sorgenfrey line is a first accounting and non-metalisable name and is therefore not homeomorphic to a topological group.
On the other hand, the Sorgenfrey line $ mathbb S $ is homeomorphic to a subset of a topological group. For example, the free topological group $ F ( mathbb S) $ more than $ mathbb S $ contains a closed topological copy of $ mathbb S $. But $ F ( mathbb S) $ also contains a topological copy of the square $ mathbb S times mathbb S $ and so $ F ( mathbb S) $ contains an unobtrusive discrete subspace. Is this situation typical?
Problem. Let $ G $ to be a topological group containing a topological copy of the Sorgenfrey line. Is $ G $ does it necessarily contain an unobtrusive discrete subspace?