Let be representatives of the conjugacy classes of a finite group and suppose that these elements commute pairwise. Prove that is abelian.
Let act on itself by conjugation. Note that, in particular, for all under this action. Let ; then for some and . Thus for each (since stabilizes each ), and more generally, for all . So for each . Since is finite, by this previous exercise, must not be a proper subgroup; that is, for each .
Now let , with and . Now . Thus is abelian.