**Problem**. Let be a finite group and a system of representatives for the conjugacy classes of . Prove that if the pairwise commute, then is abelian.

**Solution**. We will denote by the centralizer of in Suppose that is not abelian. Then and we may assume that for some Since the pairwise commute, we have for all and and hence for all Thus for all So by the class equation which gives us the contradiction

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