With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ continuous?



Existence of infinite groups that are too reluctant to be topological
There is a large literature about this, see "non-topologizable 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](http://arxiv.org/abs/math/0603513) for references) around 1980.


An observation is that for a group $G$, we have the equivalence between:


* the only group topologies on $G$ are the discrete and the indiscrete ones
* $G$ is simple and non-topologizable.


Shelah's example being simple, it therefore answers your question.


