On groups having a $p$-constant character

LetGbe a finite group, andpa prime number; a character ofGis calledp-constantif it takes a constant value on all the elements ofGwhose order is divisible byp. This is ageneralization of the very important concept of characters ofp-defect zero. In this paper, wecharacterize the finitep-solvable groups having a faithful irreducible character that isp-constantand not ofp-defect zero, and we will show that a non-p-solvable group with this property is analmost-simple group.