On zeros of characters of finite groups

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 $G$.