有限可解群的Brauer特征标表的一个注记

A Note on the Brauer Character Table of Finite Solvable Groups

Masahiko Miyamoto证明了如果A是有限群G的一个初等交换的正规q-子群,Q是G的一个西罗q-子群,那么G的所有不可约特征标都不会零化Z(Q)∩A.本文将该结果推广到Brauer特征标上,证明了若x∈Z(Q)∩Oq(G)是G的q阶元素,那么G的所有不可约p-Brauer特征标都不能零化它,其中p≠q.此外,得到对于非p-群的有限可解群,其Brauer特征标表必有一非平凡的列不取零值.

Masahiko Miyamoto has proved that if A is an elementary abelian normal q-subgroup of a finite group G and Q is a Sylow q-subgroup of G,then no irreducible character of G vanish on any element of Z（Q） ∩A. In this paper,we extend it to Brauer charac- ters and show that if x∈ Z（Q） ∩ Oq （G） of order q, then no p-Brauer characters can vanish it, where p≠q. Moreover, we obtain that for a finite solvable group which is not a p-group,its Brauer character table has a nontrivial column without zero value.