%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