For an irreducible complex character χ of the finite group G, let $\pi (\chi )$ denote the set of prime divisors of the degree $\chi (1)$ of χ. Denote then by $\rho (G)$ the union of all the sets $\pi (\chi )$ and by $\sigma (G)$ the largest value of $|\pi (\chi )|$, as χ runs in $\mathrm{Irr}(G)$. The ρ-σ conjecture, formulated by Bertram Huppert in the 80's, predicts that $|\rho (G)|\le 3\sigma (G)$ always holds, whereas $|\rho (G)|\le 2\sigma (G)$ 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 $|\rho (G)|\le 2\sigma (G)+1$ 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 $\sigma (G)\le 5$, but it is false in general if $\sigma (G)\ge 6$. Instead, we establish that $|\rho (G)|\le 3\sigma (G)-4$ holds for every finite group with a trivial Fitting subgroup and with $\sigma (G)\ge 6$ (this being the right, best possible bound). Also, we improve the up-to-date best bound for the solvable case, showing that we have $|\rho (G)|\le 3\sigma (G)$ 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.

对于有限群G的不可约复字符χ，令$\pi (\chi )$ 表示度数的素因数集合 $\chi (1)$的χ。然后用$\rho (G)$ 所有集合的并集 $\pi (\chi )$ 并由 $\sigma (G)$ 的最大值 $|\pi (\chi )|$，当χ运行$\mathrm{红外线}(G)$. 由 Bertram Huppert 在 80 年代提出的ρ-σ 猜想预测$|\rho (G)|\le 3\sigma (G)$ 总是成立，而 $|\rho (G)|\le 2\sigma (G)$如果G可解，则成立；此外，O. Manz 和 TR Wolf 在一般情况下提出了猜想的“加强”形式，询问是否$|\rho (G)|\le 2\sigma (G)+1$对每个有限群G 都成立。在本文中，我们研究了具有平凡拟合子群的有限群类的强化ρ - σ猜想：在这种情况下，我们证明该猜想是正确的，前提是$\sigma (G)\le 5$，但一般来说这是错误的，如果 $\sigma (G)\ge 6$. 相反，我们确定$|\rho (G)|\le 3\sigma (G)-4$ 对每个具有平凡拟合子群的有限群成立，并且 $\sigma (G)\ge 6$（这是正确的，最好的界限）。此外，我们改进了可解情况的最新最佳界限，表明我们有$|\rho (G)|\le 3\sigma (G)$每当G属于一个包含所有有限可解群的特定类时，我们改进了 [18] 中针对一般情况获得的最新最佳界限。