# A Problem in Group Theory. [duplicate]

How can I prove following problem in abstract algebra?

Let $G$ is a finite non-abelian group. show that there exist elements $a,g,h\in G$ such that $g\neq h, h=aga^{-1}$ and $gh=hg$.

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## marked as duplicate by T. Bongers, some1.new4u, Shuchang, Branimir Ćaćić, Erick WongDec 1 '13 at 7:09

@ChrisK No, because if you apply $h=aga^{-1}$ you get $g=h$. –  Jack M Nov 10 '13 at 15:31
Hint: I think using Conjugacy class equation: $$\displaystyle \left|{G}\right| = \left|{Z \left({G}\right)}\right| + \sum_{x_j\notin Z(G)} \left[{G : C_G \left({x_j}\right)}\right]$$ is effective here.