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 $ā (eā )x\in G$, $|C_G(x)|=\frac{|G|}{2}$. Am I on the right way? Any hints will be appreciated. Thanks.