Let be a conjugacy class in the finite group . Let be a ring with 1.
- Prove that the element is in the center of the group ring . [Hint: Check that for all .]
- Let be the conjugacy classes of , and for each , let be the sum of the elements in (as described in part 1). Prove that an element is in the center if and only if for some elements .
- Let . Note that conjugation by permutes the elements of , so that (as an element of ) we have . Thus for all . Then for all , we see the following.
= = = = = = = =
- First we show that . Let . Then we have the following.
= = = = = = = = =
Now let . First, let be arbitrary. By examining each coefficient of , we see that for all . Now recall that acts transitively (by conjugation) on each of its conjugacy classes. If is a conjugacy class of and , then we have for some . The coefficient of in is on one hand and on the other, so that in fact . In fact the coefficient of each is , and we have .