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.