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.
Manuel Dominguez Wade, P. (2017). $G$-Weights and $p$-Local Rank. Journal of Algebra and Related Topics, 5(2), 1-12. doi: 10.22124/jart.2017.2711
MLA
P. Manuel Dominguez Wade. "$G$-Weights and $p$-Local Rank". Journal of Algebra and Related Topics, 5, 2, 2017, 1-12. doi: 10.22124/jart.2017.2711
HARVARD
Manuel Dominguez Wade, P. (2017). '$G$-Weights and $p$-Local Rank', Journal of Algebra and Related Topics, 5(2), pp. 1-12. doi: 10.22124/jart.2017.2711
VANCOUVER
Manuel Dominguez Wade, P. $G$-Weights and $p$-Local Rank. Journal of Algebra and Related Topics, 2017; 5(2): 1-12. doi: 10.22124/jart.2017.2711
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