With ZFC, is there an infinite group $G$ such that there is no nontrivial nondiscrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{1}$ continuous?
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There is a large literature about this, see "nontopologizable groups". These are, by definition, groups for which the only Hausdorff group topology is discrete. There are various examples, the first of which were obtained by Olshanskii and Shelah (see here for references) around 1980. An observation is that for a group $G$, we have the equivalence between:
Shelah's example being simple, it therefore answers your question. 

