Character triples and relative defect zero characters

Junwei Zhang, Lizhong Wang, Ping Jin
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引用次数: 0

Abstract

Given a character triple $(G,N,\theta)$, which means that $G$ is a finite group with $N \vartriangleleft G$ and $\theta\in{\rm Irr}(N)$ is $G$-invariant, we introduce the notion of a $\pi$-quasi extension of $\theta$ to $G$ where $\pi$ is the set of primes dividing the order of the cohomology element $[\theta]_{G/N}\in H^2(G/N,\mathbb{C}^\times)$ associated with the character triple, and then establish the uniqueness of such an extension in the normalized case. As an application, we use the $\pi$-quasi extension of $\theta$ to construct a bijection from the set of $\pi$-defect zero characters of $G/N$ onto the set of relative $\pi$-defect zero characters of $G$ over $\theta$. Our results generalize the related theorems of M. Murai and of G. Navarro.
字符三元组和相对缺陷零字符
给定一个特征三元组 $(G,N,\theta)$,这意味着 $G$ 是一个有限群,有 $N \vartriangleleft G$,并且 $\theta\in\{rm Irr}(N)$是 $G$ 不变的、我们引入了$\theta$到$G$的$\pi$-准扩展的概念,其中$\pi$是除以H^2(G/N,\mathbb{C}^\times)$中与特征三元组相关的同调元素$[\theta]_{G/N}\的阶的素集,然后建立了这种扩展在规范化情况下的唯一性。作为应用,我们使用$theta$的$\pi$-准扩展来构造一个从$G/N$的$\pi$-缺陷零字符集到$G$在$theta$上的相对$\pi$-缺陷零字符集的双射。我们的结果概括了 M. Murai 和 G. Navarro 的相关定理。
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