Character triples and relative defect zero characters

TL;DR

This paper introduces the concept of $ ext{pi}$-quasi extension for characters in finite groups, establishing their uniqueness and using them to relate $ ext{pi}$-defect zero characters between groups and their quotients.

Contribution

It defines $ ext{pi}$-quasi extensions of characters in finite groups and proves their uniqueness, extending previous theorems by Murai and Navarro.

Findings

Established the uniqueness of $ ext{pi}$-quasi extensions in normalized cases.

Constructed a bijection between $ ext{pi}$-defect zero characters of $G/N$ and relative $ ext{pi}$-defect zero characters of $G$.

Generalized earlier results on defect zero characters in finite group theory.

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.