Useful sufficient conditions for a topological space to be the underlying space of a topological group?
Solution 1:
In this short paper we have a counterexample for an even weaker statement (it admits no left topological group structure). There are also consistent examples (under CH) that are S-spaces etc. Note that any first countable compact zero-dimensional homogeneous space that is not metrisable is a counterexample, as a first countable topological group is metrisable.