关于单模的顶点和核的一点注记

Abstract

设G为有限群,k是特征为p的代数闭域(p>0).另设S是单kG-模,V(S)是S的一个顶点,Ker(S)是S的核.在本文中,若Op′(G)■Z(G)且每个属于主p-块的单kG-模S均有V(S)■Ker(S),则对每个x∈G,令Q=P∩Px,G中存在一个包含Op′(G)的正规子群H,满足Q∈Sylp(H)且|NH(Q)/Q|=|Op′(G)|.另外,设B为G的一个p-块,得到了B为p-根块的一个充分条件.

Let Gbe a finite group and ka field of characteristic p>0.And let Sbe a simple kG-module and V(S)a vertex of S.Also denote by Ker(S)the kernel of S.In this note,we give a group theoretical characterization of Gif V(S)■Ker(S)for any simple kGmodule Sin the principal p-block of G.In addition,we obtain that a sufficient condition for p-radical blocks of G.