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

For a character χ of a finite group G, the number χc(1)=[G:kerχ]χ(1) is called the codegree of χ. Let N be a normal subgroup of G and set Irr(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|, pq 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.