Character triples and relative defect zero characters

Abstract

Given a character triple $(G,N,\hat{I}¸)$, which means that $G$ is a finite group with $N⊲G$ and $\hat{I}¸\in \mathrm{Irr}(N)$ is $G$-invariant, we introduce the notion of a $\ddot{I}€$-quasi extension of $\hat{I}¸$ to $G$ where $\ddot{I}€$ is the set of primes dividing the order of the cohomology element $[\hat{I}¸{]}_{G/N}\in H^2(G/N,{\mathbb{C}}^{\times })$ associated with the character triple, and then establish the uniqueness of such an extension in the normalized case. As an application, we use the $\ddot{I}€$-quasi extension of $\hat{I}¸$ to construct a bijection from the set of $\ddot{I}€$-defect zero characters of $G/N$ onto the set of relative $\ddot{I}€$-defect zero characters of $G$ over $\hat{I}¸$. Our results generalize the related theorems of M. Murai and of G. Navarro.