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- W3147199799 abstract "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 $|rho(G)|leq 3sigma(G)$ always holds, whereas $|rho(G)|leq 2sigma(G)$ holds if $G$ is solvable; moreover, O. Manz and T.R. Wolf proposed a form of the conjecture in the general case, asking whether $|rho(G)|leq 2sigma(G)+1$ 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 $|rho(G)|leq 3sigma(G)-4$ 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 $|rho(G)|leq 3sigma(G)$ whenever $G$ belongs to one particular class including all the finite solvable groups." @default.
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- W3147199799 date "2021-03-26" @default.
- W3147199799 modified "2023-09-27" @default.
- W3147199799 title "On Huppert's Rho-Sigma Conjecture." @default.
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