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Paper   IPM / M / 16692
School of Mathematics
  Title:   On Huppert Rho-Sigma conjecture
  Author(s):  Zeinab Akhlaghi (Joint with S. Dolfi and E. Pacifici)
  Status:   Published
  Journal: J. Algebra
  Year:  2021
  Pages:   DOI: 10.1016/j.jalgebra.2021.06.038
  Supported by:  IPM
  Abstract:
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 80�??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 �??strengthened�?� 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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