Homological properties of the endomorphism rings of certain permutation modules (Q914826)

Let G be a finite group and let F be a field of prime characteristic p. The author studies homological properties of the endomorphism ring \(\Lambda\) of the FG-module \(M=\oplus F(G/H)\), summed over all subgroups H of G. One of the main results (Theorem A) asserts that if G is not a \(p'\)-group, then the finitistic dimension of the ring \(\Lambda\) is equal to \(1+\sup \{rank H/\Phi (H)\}\) where H runs over all p-subgroups of G and \(\Phi\) (H) denotes the Frattini subgroup of H.