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On the minimal norm of a non‐regular generalized character of an arbitrary finite group

We prove that, for any generalized character θ, other than a multiple of the regular character, of any finite group G, we have ∑g∊G#|θ(g)|2 ⩾ |G|χ(1) − 1, where χ is an irreducible character of maximal degree of G and G# is the set of non‐identity elements of G. We also determine exactly when equality is achieved.