Groups with some cube-free irreducible character co-degrees

Abstract

For a character χ of a finite group G, the number is called the co-degree of χ. Let be an integer and denote the set of irreducible characters whose kernels do not contain . In this paper, we show that if G is solvable and for every prime divisor p of and every , then the derived length of G is at most . Then, we classify the finite non-solvable groups with non-trivial Fitting subgroups such that the co-degrees of their irreducible characters whose kernels do not contain the Fitting subgroups are cube-free.