The prime-power hypothesis on the codegrees of irreducible characters

For a character χ of a finite group G, the number cod(χ)=G:Kerχχ(1) is called the codegree of χ. In this paper, we show that if for every non-principal irreducible characters χ and ϕ of G with cod(χ)≠cod(ϕ), the greatest common divisor of cod(χ) and cod(ϕ) is a prime-power, then either G is solvable or G/Sol(G) is isomorphic to one of the groups Alt5 or PSL2(8), where Sol(G) is the solvable radical of the group G.