On Huppert's rho-sigma conjecture

[摘要] For an irreducible complex character x 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 vertical bar pi(chi)vertical bar, as chi runs in Irr(G). The rho-sigma conjecture, formulated by Bertram Huppert in the 80's, predicts that vertical bar rho(G)vertical bar <= 3 sigma(G) always holds, whereas vertical bar rho(G)vertical bar <= 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 vertical bar rho(G)vertical bar <= 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) <= 5, but it is false in general if cr(G)> 6. Instead, we establish that vertical bar rho(G)vertical bar <= 3 sigma(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 vertical bar rho(G)vertical bar <= 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. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-11-15

[关键词] Finite groups;Character degrees