Which elements of a finite group are non-vanishing?

Abstract

‎Let $G$ be a finite group‎. ‎An element $gin G$ is called non-vanishing‎, ‎if for‎ ‎every irreducible complex character $chi$ of $G$‎, ‎$chi(g)neq 0$‎. ‎The bi-Cayley graph ${rm BCay}(G,T)$ of $G$ with respect to a subset $Tsubseteq G$‎, ‎is an undirected graph with‎ ‎vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(tx,2)}mid xin G‎, ‎ tin T}$‎. ‎Let ${rm nv}(G)$ be the set‎ ‎of all non-vanishing elements of a finite group $G$‎. ‎We show that $gin nv(G)$ if and only if the adjacency matrix of ${rm BCay}(G,T)$‎, ‎where $T={rm Cl}(g)$ is the‎ ‎conjugacy class of $g$‎, ‎is non-singular‎. ‎We prove that ‎if the commutator subgroup of $G$ has prime order $p$‎, ‎then‎ ‎(1) $gin {rm nv}(G)$ if and only if $|Cl(g)|<p$, <br />‎(2) if $p$ is the smallest prime divisor of $|G|$‎, ‎then ${rm nv}(G)=Z(G)$‎. <br />‎‎Also we show that‎ <br />(a) if ${rm Cl}(g)={g,h}$‎, ‎then $gin {rm nv}(G)$ if and only if $gh^{-1}$ has odd order‎, <br />(b) if $|{rm Cl}(g)|in {2,3}$ and $({rm ord}(g),6)=1$‎, ‎then $gin {rm nv}(G)$‎.