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

	$G$-Weights and $p$-Local Rank
Journal of Algebra and Related Topics
مقاله 1، دوره 5، شماره 2، اسفند 2017، صفحه 1-12 اصل مقاله (307.6 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22124/jart.2017.2711
نویسنده
P. Manuel Dominguez Wade^*
Department of Mathematics, Matanzas University, Matanzas, Cuba
چکیده
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.
کلیدواژه‌ها
Radical vertex؛ $G$-weight؛ $p$-local rank

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