WHICH ELEMENTS OF A FINITE GROUP ARE NON-VANISHING?

Let G be a finite group. An element g is an element of G is called non vanishing, if for every irreducible complex character chi of G, chi(g) not equal 0. The bi-Cayley graph BCay(G, T) of G with respect to a subset T subset of G, is an undirected graph with vertex set G x {1, 2} and edge set {{(x, 1), (tx, 2)} x is an element of G, t is an element of T}. Let nv(G) be the set of all non -vanishing elements of a finite group G. We show that g is an element of 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 is an element of nv(G) if and only if vertical bar Cl(g)vertical bar < p, (2) if p is the smallest prime divisor of vertical bar G vertical bar, then nv(G) = Z(G). Also we show that (a) if Cl(g) = {g, h}, then g is an element of nv(G) if and only if gh(-1) has odd order, (b) if vertical bar Cl(g)vertical bar is an element of {2, 3} and (o(g), 6) = 1, then g is an element of nv(G).