%0 Journal Article %T $G$-Weights and $p$-Local Rank %J Journal of Algebra and Related Topics %I University of Guilan %Z 2345-3931 %A Manuel Dominguez Wade, P. %D 2017 %\ 12/01/2017 %V 5 %N 2 %P 1-12 %! $G$-Weights and $p$-Local Rank %K Radical vertex %K $G$-weight %K $p$-local rank %R 10.22124/jart.2017.2711 %X 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. %U https://jart.guilan.ac.ir/article_2711_8f0e0342d1bad2b4ab66eb7767948d0e.pdf