TITLE: Groups where no elements commute except for the trivial cases
QUESTION [11 upvotes]: Let $G$ be a group. Write $e$ for its neutral element and write $\langle g\rangle$ for the subgroup generated by an element $g \in G$. Assume that $G$ has the following properties:

For all $g\in G\setminus\{e\}$ and $h\in G\setminus \langle g \rangle$ we have $gh \neq hg$.
Property 1. is non-vacuous (as it would be e.g. for $G=\{e\}$).

Do such groups exist? If so, do they have any interesting/important properties? Note that this is a follow-up to this very similar question in response to one of the comments there.

REPLY [9 votes]: To provide contrast to Andreas Caranti's answer, a Tarski monster $p$-group is an example of an infinite group with these properties.
If $G$ is a Tarski monster $p$-group then (by definition) every proper nontrivial subgroup is cyclic of order $p$. The centralizer of any nontrivial element $g$ is a proper subgroup (since $G$ has trivial center), and thus must be $\langle g\rangle$.