G-Weights and p-Local Rank

Let k be field of characteristic p, andlet 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 permutationmodule 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.

Keywords:

Radical vertex , G-weight , p-local rank