On Huppert's Rho-Sigma Conjecture

DOI10.1016/J.JALGEBRA.2021.06.038arXiv2103.14316WikidataQ123027278 ScholiaQ123027278MaRDI QIDQ6363808

Z. Akhlaghi, Silvio Dolfi, Emanuele Pacifici

Publication date: 26 March 2021

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 $ho(G)$ the union of all the sets $pi(chi)$ and by $sigma(G)$ the largest value of $|pi(chi)|$, as $chi$ runs in $mIrr(G)$. The $ho$-$sigma$ conjecture, formulated by Bertram Huppert in the 80's, predicts that $|ho(G)|leq3sigma(G)$ always holds, whereas $|ho(G)|leq2sigma(G)$ 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 $|ho(G)|leq2sigma(G)+1$ is true for every finite group $G$. In this paper we study the strengthened $ho$-$sigma$ conjecture for the class of finite groups having a trivial Fitting subgroup: in this context, we prove that the conjecture is true provided $sigma(G)leq5$, but it is false in general if $sigma(G)geq6$. Instead, we establish that $|ho(G)|leq3sigma(G)-4$ holds for every finite group with a trivial Fitting subgroup and with $sigma(G)geq6$ (this being the right, best possible bound). Also, we improve the up-to-date best bound for the solvable case, showing that we have $|ho(G)|leq3sigma(G)$ whenever $G$ belongs to one particular class including all the finite solvable groups.

Mathematics Subject Classification ID

Ordinary representations and characters (20C15)

This page was built for publication: On Huppert's Rho-Sigma Conjecture

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6363808)