On Huppert's Rho-Sigma Conjecture

For an irreducible complex character $\chi$ of the finite group $G$, let $\pi(\chi)$ denote the set of prime divisors of the degree $\chi(1)$ of $\chi$. Denote then by $\rho(G)$ the union of all the sets $\pi(\chi)$ and by $\sigma(G)$ the largest value of $|\pi(\chi)|$, as $\chi$ runs in ${\rm{Irr}}(G)$. The $\rho$-$\sigma$ conjecture, formulated by Bertram Huppert in the 80's, predicts that $|\rho(G)|\leq 3\sigma(G)$ always holds, whereas $|\rho(G)|\leq 2\sigma(G)$ 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 $|\rho(G)|\leq 2\sigma(G)+1$ is true for every finite group $G$. In this paper we study the strengthened $\rho$-$\sigma$ conjecture for the class of finite groups having a trivial Fitting subgroup: in this context, we prove that the conjecture is true provided $\sigma(G)\leq 5$, but it is false in general if $\sigma(G)\geq 6$. Instead, we establish that $|\rho(G)|\leq 3\sigma(G)-4$ holds for every finite group with a trivial Fitting subgroup and with $\sigma(G)\geq 6$ (this being the right, best possible bound). Also, we improve the up-to-date best bound for the solvable case, showing that we have $|\rho(G)|\leq 3\sigma(G)$ whenever $G$ belongs to one particular class including all the finite solvable groups.