p-parts of co-degrees of irreducible characters

For a character χ of a finite group G , the co-degree of χ is χc (1) = [G :kerχ] χ(1) . Let p be a prime and let e be a positive integer. In this paper, we first show that if G is a p-solvable group such that pe+1 χc (1), for every irreducible character χ of G , then the p-length of G is not greater than e. Next, we study the finite groups satisfying the condition that p2 does not divide the co-degrees of their irreducible characters. Mathematical subject classification (2010). 20C15, 20D10, 20D05. Manuscript received 22nd April 2020, revised and accepted 24th November 2020.