A note on Huppert's \(\rho\)-\(\sigma\) conjecture: an improvement on a result by Casolo and Dolfi. (Q2874695)

Let \(G\) be a finite group. For \(n\in\mathbb N\) let \(\sigma(n)\) denote the number of distinct primes dividing \(n\), and let \(\sigma(G)=\max\{\sigma(\chi(1))\mid\chi\in\text{Irr}(G)\}\) be the maximum number of primes occurring in an irreducible (complex) character degree of \(G\). Let \(\rho(G)\) be the set of primes dividing some irreducible character degree of \(G\). Huppert's \(\rho\)-\(\sigma\)-conjecture asserts that \(|\rho(G)|\) is bounded by some function depending only on \(\sigma(G)\), and more specifically, \(|\rho(G)|\leq C\sigma(G)\) with \(C=3\) in general and \(C=2\) for solvable \(G\).NEWLINENEWLINE The best known bounds to date are \(|\rho(G)|\leq 3\sigma(G)+2\) for solvable \(G\) [\textit{O. Manz} and \textit{T. R. Wolf}, Ill. J. Math. 37, No. 4, 652-665 (1993; Zbl 0832.20005)] and \(|\rho(G)|\leq 7\sigma(G)\) in general [\textit{C. Casolo} and \textit{S. Dolfi}, J. Group Theory 10, No. 5, 571-583 (2007; Zbl 1127.20006)].NEWLINENEWLINE In the paper under review, the latter general result is improved as follows: \(|\rho(G)|\leq 6\sigma(G)+1\). This is obtained by improving Proposition 3 in the Casolo-Dolfi paper [op. cit.], and this improvement rests on a result on the covering number for nonabelian simple groups: It is 2 except for \(J_1\) and \(\text{PSL}(2,q)\) for which it is 3. (Here the covering number of the simple group \(G\) is the smallest cardinality of a set of character degrees of \(G\) such that every prime dividing \(|G|\) divides at least one character degree in that set.)