On groups having a p-constant character

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.