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Let G be a finite group, and p a prime number; a character of G is called p-constant if it takes a constant value on all the elements of G whose order is divisible by p. This is a generalization of the very important concept of characters of p-defect zero. In this paper, we characterize the finite p-solvable groups having a faithful irreducible character that is p-constant and not of p-defect zero, and we will show that a non-p-solvable group with this property is an almost-simple group.

On groups having a 𝑝-constant character