Which elements of a finite group are non-vanishing?

Document Type : Research Paper


1 Department of‎ ‎Mathematical Sciences, Isfahan University‎ ‎of Technology‎, ‎P‎.‎O‎. ‎Box 84156-83111, Isfahan‎, ‎Iran.

2 Department of‎ ‎Mathematical Sciences, Isfahan University‎ ‎of Technology‎, ‎P‎.‎O‎. ‎Box 84156-838111, Isfahan‎, ‎Iran.


‎Let $G$ be a finite group‎. ‎An element $g\in G$ 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 $T\subseteq G$‎, ‎is an undirected graph with‎ ‎vertex set $G\times\{1,2\}$ and edge set $\{\{(x,1),(tx,2)\}\mid x\in G‎, ‎\ t\in T\}$‎. ‎Let ${\rm nv}(G)$ be the set‎ ‎of all non-vanishing elements of a finite group $G$‎. ‎We show that $g\in nv(G)$ 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) $g\in {\rm nv}(G)$ if and only if $|Cl(g)|<p$,
‎(2) if $p$ is the smallest prime divisor of $|G|$‎, ‎then ${\rm nv}(G)=Z(G)$‎.
‎‎Also we show that‎
(a) if ${\rm Cl}(g)=\{g,h\}$‎, ‎then $g\in {\rm nv}(G)$ if and only if $gh^{-1}$ has odd order‎,
(b) if $|{\rm Cl}(g)|\in \{2,3\}$ and $({\rm ord}(g),6)=1$‎, ‎then $g\in {\rm nv}(G)$‎.


Main Subjects

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