The Fitting subgroup, p ‐length, derived length and character table

For a character χ of a finite group G , the numberχ c ( 1 ) = [ G : ker χ ] χ ( 1 )is called the codegree of χ. Let N be a normal subgroup of G and setIrr ( G | N ) = Irr ( G ) − Irr ( G / N ) . Let p be a prime. In this paper, we first show that if for two distinct prime divisors p and q of | N | , p q divides none of the codegrees of elements of Irr ( G | N ) , then Fit ( N ) ≠ { 1 } and N is either p ‐solvable or q ‐solvable. Next, we classify the finite groups with exactly one irreducible character of the codegree divisible by p and, also finite groups whose codegrees of irreducible characters which are divisible by p are equal. Then, we prove that p ‐length of a finite p ‐solvable group is not greater than the number of the distinct codegrees of its irreducible characters which are divisible by p . Finally, we consider the case when the codegree of every element of Irr ( G | N ) is square‐free.