G -Weights and p -Local Rank

Let k be field of characteristic p , and let G be any finite group with splitting field k . Assume that B is a p -block of G .In this paper, we introduce the notion of radical B -chain C B , and we show that the p -local rank of B is equals the length of C B . Moreover, we prove that the vertex of a simple kG -module S is radical if and only if it has the same vertex of the unique direct summand, up to isomorphism, of the Sylow permutation module whose radical quotient is isomorphic to S . Finally, we prove the vertices of certain direct summands of the Sylow permutation module are bounds for the vertices of simple kG -modules.