Let be the representatives of the conjugacy classes of the finite group G and assume that these elements pairwise commute. Prove that G is abelian.

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- Mar 6th 2011, 04:13 AMabhishekkgpconjugacy classes
Let be the representatives of the conjugacy classes of the finite group G and assume that these elements pairwise commute. Prove that G is abelian.

- Mar 6th 2011, 08:21 AMTinyboss
Did you try anything or have any ideas yet? How far did you get?

- Mar 7th 2011, 12:10 AMabhishekkgp
- Mar 7th 2011, 09:39 AMtopspin1617
Hmm.. well that's not the way to go. Just because representatives of classes commute, doesn't mean that all elements between the classes commute. For example, consider the classes . There must be another way..

EDIT: Try this.

First, we need the fact that, for any finite group and subgroup , .

Now, we let act on itself by conjugation. Since it is assumed that the commute pairwise, we have for all . That is, for each , (the stabilizer of under this group action).

Let be arbitrary. Then there exist such that . Hence for each .

We have shown that, for any , we can find so that . It follows that (for each ).

By the claim above, since is finite, this forces for each . That is, the conjugacy class of is . We know that this means and, since these elements form a complete set of representatives for the conjugacy classes of , , and the group is abelian. - Mar 7th 2011, 07:09 PMabhishekkgp
- Mar 7th 2011, 07:31 PMabhishekkgp