TITLE: Is there any clasification of such groups?
QUESTION [8 upvotes]: Let $G$ be a finite group such that for all $x\neq e$, 
$$C_G(x)=\langle x\rangle$$ 
Is there any classification of such groups ?

REPLY [3 votes]: Same basic idea as in the answer by Jack.
First of all, it is immediate that every nontrivial element of $G$ must have prime order. Furthermore, $p^2$ does not divide the order of $G$ for any prime $p$, as otherwise an element of order $p$ would be contained in a subgroup of order $p^2$ (which is abelian). 
Thus $G$ has squarefree order. It is a basic fact about groups of squarefree order that $G$ has a normal $p$-Sylow $P$, where $p$ is the largest prime dividing $|G|$ (you could prove this with Burnside's normal complement theorem). Then $\operatorname{Aut}(P)$ is abelian, so $N_G(P)/C_G(P) = G/P$ is abelian. Since $|G|$ is squarefree, $G/P$ is cyclic. Because every nontrivial element of $G$ has prime order, this is true for $G/P$ as well. Hence either $G/P$ is trivial or has prime order.
It follows then that either $G$ has prime order, or is nonabelian of order $pq$ for distinct primes $p$ and $q$.