Does there exist a group $G$ such that $G$ has no topology on it such that $G$ is a topological group apart from the (in)discrete topology (or other such trivalish topologies)? I am asking as interested in the general methods that one construct a topological group from a group.
I am quite interested in how the problem changes if $G$ is infinite or finite.
You can find this discussed very nicely in the section about Markov's problems in these notes by Dikran Dikranjan.
In particular, there do exists groups which have no non-discrete compatible topologies; the notes include the examples of Adian groups,which are countable, and noncountable examples due to Shelah. A nice result is that a group with infinite center has some non-discrete Hausdorff topology.