$G$-Weights and $p$-Local Rank

Document Type: Research Paper

Author

Department of Mathematics, Matanzas University, Matanzas, Cuba

Abstract

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.

Keywords