Abstract

Let G be a finite group. For a character χ of G, the number $\text{cod}(\chi )=\frac{[G:\text{ker}\chi ]}{\chi (1)}$ is called the co-degree of χ. Let N be a non-trivial normal subgroup of G and set $\text{Irr}(G|N)=\text{Irr}(G)-\text{Irr}(G/N).$ In this paper, we show that if for every $\chi ,\psi \in \text{Irr}(G|N)$ with $\text{cod}(\chi )\ne \text{cod}(\psi ),$ the greatest common divisor of $\text{cod}(\chi )$ and $\text{cod}(\psi )$ is either 1 or a prime, then N is solvable.

中文翻译：

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关于不可约字符的协度的单素假设

摘要

令G为有限群。用于字符χ的ģ，数$\text{鳕鱼}(\chi )=\frac{[G：\text{克}\chi ]}{\chi (1)}$称为χ的协度。设N是G 的一个非平凡正规子群，并且集合$\text{红外线}(G|N)=\text{红外线}(G)-\text{红外线}(G/N).$ 在本文中，我们证明如果对于每个 $\chi ,\psi \in \text{红外线}(G|N)$ 和 $\text{鳕鱼}(\chi )\ne \text{鳕鱼}(\psi ),$ 的最大公约数 $\text{鳕鱼}(\chi )$ 和 $\text{鳕鱼}(\psi )$是 1 或素数，则N是可解的。