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- Title
WHICH ELEMENTS OF A FINITE GROUP ARE NON-VANISHING?
- Authors
AREZOOMAND, M.; TAERI, B.
- Abstract
Let G be a finite group. An element g∈G is called non-vanishing, if for every irreducible complex character χ of G , χ(g)≠0. The bi-Cayley graph BCay(G,T) of G with respect to a subset T⊆G, is an undirected graph with vertex set G×{1,2} and edge set {{(x,1),(tx,2)}∣x∈G, t∈T}. Let nv(G) be the set of all non-vanishing elements of a finite group G . We show that g ∈ nv(G) if and only if the adjacency matrix of BCay(G,T), where T=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 ∈ nv(G) if and only if|Cl(g)|<p, (2) if p is the smallest prime divisor of |G|, then nv(G)=Z(G). Also we show that (a) if Cl(g)={g,h}, then g ∈ nv(G) if and only if gh −1 has odd order, (b) if |Cl(g)|∈{2,3} and (o(g),6)=1, then g ∈ nv(G).
- Publication
Bulletin of the Iranian Mathematical Society, 2016, Vol 42, Issue 5, p1097
- ISSN
1018-6301
- Publication type
Academic Journal