On a subalgebra of the centre of a group ring

Let p be a prime, G a finite group with pG and F a field of characteristic p. Let C1,…,Cr be the p-regular conjugacy classes in G, the class sums in FG and the F-subspace of the centre Z(FG) of FG generated by the p-regular class sums. We show that is a subalgebra of Z(FG), if G has abelian Sylow-p-subgroups. Furthermore we classify some minimal groups, where is not an algebra.