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NOTE ON AN INEQUALITY OF BURNSIDE

A well-known property of irreducible characters of finite groups is the fact, proved by Burnside, that if G is a finite group and % : G −→ GL(E) is an irreducible complex representation of dimension dim(%) > 2, then there exists some g ∈ G such that χ%(g) = Tr %(g) is zero. A common argument to prove this relies on the following result (also due to Burnside) on representations of finite cyclic groups: