Is there any finite non trivial Group with this property? I was asked to have a look at a problem:

There is no a finite non-trivial group $G$ that all its non-trivial elements can be commuted with exactly half elements of group .

For the first step, I saw I could not prove it directly so, I assumed we have such a group $G$, finite and satisfying above property. The property led $|G|$ to have an even order because $∀ (e≠)x\in G$, $|C_G(x)|=\frac{|G|}{2}$. Am I on the right way? Any hints will be  appreciated. Thanks.

You're on the right track. Remember that $[G: C_G(x)]$ is the size of the conjugacy class of $x \in G$. Use the fact that $G$ is a disjoint union of conjugacy classes to complete your solution.
