On zeros of characters of finite groups
David Chillag
Department of Mathematics, Technion, Israel Institute of Technology, 
Haifa 32000, Israel 
Proc. Amer. Math. Soc. 127 (1999), 977-983.

Abstract. We present several results connecting the number of conjugacy 
classes of a finite group on which an irreducible character vanishes, and the 
size of some centralizer of an element. For example, we show that if $G$ is a 
finite group such that $G \ne G' \ne G''$, then $G$ has an element $x$, such 
that $|C_G(x)| \le 2m$, where $m$ is the maximal number of zeros in a row of 
the character table of $G$. Dual results connecting the number of irreducible 
characters which are zero on a fixed conjugacy class, and the degree of some 
irreducible character, are included too. For example, the dual of the
above result is the following: Let $G$ be a finite group such that 
$1 \ne Z(G) \ne Z_2(G)$; then $G$ has an irreducible character $\chi$ such 
that $\frac{|G|}{\chi^2(1)} \le 2m$, where $m$ is the maximal number of
zeros in a column of the character table of $G$.