Conjugacy class of more than average size has character value zero for some irreducible character

Statement

Suppose $G$ is a finite group and $c$ is a conjugacy class in $G$ such that $c$ has size more than the average conjugacy class size, i.e., $|G|/|c|$ is less than the number of conjugacy classes in $G$.

Then, there exists an irreducible representation of $G$ over the complex numbers with character $\chi$ such that $\chi(g) = 0$ for some $g \in c$.

Related facts

• Central implies image under every irreducible representation is scalar
• Irreducible character of degree greater than one takes value zero on some conjugacy class
• Zero-or-scalar lemma