Let be field of characteristic , and let be any finite group with splitting field . Assume that is a -block of . In this paper, we introduce the notion of radical -chain , and we show that the -local rank of is equals the length of . Moreover, we prove that the vertex of a simple -module 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 . Finally, we prove the vertices of certain direct summands of the Sylow permutation module are bounds for the vertices of simple -modules.
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