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

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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.

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