π-Hall子群的线性特征标的诱导

Induced Characters from Linear Characters of a π-Hall Subgroup

诱导特征标研究群G的特征标与它的子群的特征标之间的关系,其主要目的是利用G的子群已知的不可约特征标来获得G的一些不可约特征标,从而了解G的结构.McKay猜想断言:设G为任意有限群,p为任意素数,N为G的一个Sylowp-子群P在G中的正规化子,则G和N的p′-次不可约复特征标的个数恰好相等.显然N的每个p′-次不可约复特征标在P上的限制均为线性特征标.在研究G和N的p′-次不可约复特征标之间可能存在的典范对应时,Navarro于2003年在J.Alg上发表了关于Sylowp-子群P的线性特征标到N和G的诱导性质.本文利用特征标的诱导公式,通过研究群与子群的共轭类关系,将其中的Sylowp-子群替换为π-Hall子群,对Navarro文中的3个主要定理做了更进一步的推广,这同时是对McKay猜想π-形式的研究.

The concept of indueed character plays an important role in investigating the irreducible characters of G by the irreducible characters of its subgroups. If P is a Sylow p-subgroup of G and N= No, （P） .then equal numbers of the irreducible characters of G and of N have degrees not divisible by p. This is the Mekay conjecture. If θ is an irreducible character of N which is p＇-degree, then θr is a linear charaeter. The induced characters of linear characters of Sylow p-subgroup were investigated by Navarro. This paper investigated the induced characters from linear characters of nilpotent π-Hall subgroups. The main results generalizes the theorem which was proved by Navarro. It was a further research about Mckay conjecture of π-forms.