Non-vanishing elements and complex group algebras

Abstract

Let $G$ be a finite group, and let $\mathrm{Irr}(G)$ denote the set of irreducible complex characters of $G$. An element $x$ of $G$ is said to be vanishing, if for some $\ddot{I}‡$ in $\mathrm{Irr}(G)$, we have $\ddot{I}‡(x)=0$. Also the element $x$ is called rational if $x$ is conjugate to $x^i$ for every integer $i$ co-prime to the order of $x$. We define the weight of $G$ as $\ddot{I}‰(G):=({\sum }_{\ddot{I}‡\in \mathrm{Irr}(G)}\ddot{I}‡(1){)}^2/|G|$. In this paper, we show that for every rational non-vanishing element $x\in G$, the order of $C_G(x)$ is at least $\ddot{I}‰(G)$.