Let G be a finite group and consider k[G] where k is a field. In the scenario where char(k) divides |G|, how can one show that the dimension of Z(k[G]/rad k[G]) is strictly less than dimension of Z(k[G])?
I'm trying to use this to show that in the case where char(k) divides |G|, the number of simple modules (up to isomorphism) is strictly less than no. of conjugacy classes of G. I'm aware of the fact that this can be answered using some knowledge of characters but I'm wondering if the above line of attack is also possible?
Remark: In the case where char(k) does not divide |G|, we have that rad(k[G]) is trivial since k[G] is semisimple and therefore Z(k[G]/rad k[G]) = Z(k[G]) and it can be shown that the number of simple modules is at most the number of conjugacy classes.
Thank you in advance for any light that can be shed on the matter! :)