On Huppert's rho-sigma conjecture

For an irreducible complex character χ of the finite group G, let π(χ) denote the set of prime divisors of the degree χ(1) of χ. Denote then by ρ(G) the union of all the sets π(χ) and by σ(G) the largest value of |π(χ)|, as χ runs in Irr(G). The ρ-σ conjecture, formulated by Bertram Huppert in the 80's, predicts that |ρ(G)|≤3σ(G) always holds, whereas |ρ(G)|≤2σ(G) holds if G is solvable; moreover, O. Manz and T.R. Wolf proposed a “strengthened” form of the conjecture in the general case, asking whether |ρ(G)|≤2σ(G)+1 is true for every finite group G. In this paper we study the strengthened ρ-σ conjecture for the class of finite groups having a trivial Fitting subgroup: in this context, we prove that the conjecture is true provided σ(G)≤5, but it is false in general if σ(G)≥6. Instead, we establish that |ρ(G)|≤3σ(G)−4 holds for every finite group with a trivial Fitting subgroup and with σ(G)≥6 (this being the right, best possible bound). Also, we improve the up-to-date best bound for the solvable case, showing that we have |ρ(G)|≤3σ(G) whenever G belongs to one particular class including all the finite solvable groups, and we improve the up-to-date best bound obtained in [18] for the general case.