通过稳定子(Stabilizer)的单项特征标

PASSING TO STABILIZER WITH MONOMIAL CHARACTERS

假设G为有限群,N为G的正规子群,χ∈Irr(G),θ∈Irr(Nχ_N),则由Clifford定理存在唯一的χ_0∈Irr(G_0).使得χ=(χ_0)~G.而(χ_0)_N=e·θ,此处e为正整数。G_θ={g∈G|θ~=θ}为G作用在θ上的稳定子。若χ是单项时,则χ_θ可能不是单项的。本文运用[4]的方法,用正规子群N的内部结构,给出χ_0是单项的充分条件。主要结果:当N是超可解时,χ_0是单项的。当N存在正规的Sylow p-子群时,且θ是p-特征标时,χ_0是单项的。

An irreducible (complex) character χ of finite group G is monomial if it is induced: χ/=ψ~G' from a linear (i.e.degree one) character of some subgroup N of G. Let N△G,χ∈Irr(G) and 0∈Irr(N|χ_x),by the Clifford's theorem,there is a unique χ(?)∈Irr(G_0),(G_0 is the stabilizer of 0 in the action of G on Irr(N)) such that (χ_0)_x is multiple of 0,and such that (χ_0)~G=χ.Also,if ψ∈Irr(G_0|0~GO) then ψ~G∈Irr (G) and ψ= (ψ~G_0) It is possible for χ and 0 to be monomial,yet χ_0 is not With the help of method of [4],the following main result are obtained: When N is supersolvable grup or N has a normal sylow p—group and 0 is p—character then χ_0 is monomial.