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.

## Comments

What is ? representatives of the the non-trivial orbits of the action of conjugation?

Do you mean little ? This is just an element of which conjugates our arbitrary into one of the .

Ah- I understand the question now. should be . I’m a little slow. 🙂

This is trivial, but isn’t the notation awkward? it will be better .

I was probably thinking of as a function that takes group elements to their conjugacy class representatives.