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For an irreducible complex character X of the finite group G, let Pi(G) denote the set of
prime divisors of the degree x(1) of G. Denote then by p(G) the union of all the sets pi(x) and by
sigma(G) the largest value of |Pi(x)|, as x runs in Irr(G). The rho-sigma conjecture, formulated by Bertram
Huppert in the 80ï¿½??s, predicts that |p(G)| <= 3sigma(G) always holds, whereas |p(G)|< 2sigma(G) holds if G is
solvable; moreover, O. Manz and T.R. Wolf proposed a ï¿½??strengthenedï¿½?ï¿½ form of the conjecture in the
general case, asking whether |p(G)| < 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) < 5, but it is false in general if
sigma(G) > 6. Instead, we establish that |p(G)| < 3sigma(G) - 4 holds for every finite group with a trivial
Fitting subgroup and with sigma(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 |p(G)| <= 3sigma(G) whenever G belongs
to one particular class including all the finite solvable groups.
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