On Huppert's \(\rho\)-\(\sigma\) conjecture
Let \(G\) be a finite group. Then Huppert's \(\rho\)-\(\sigma\) conjecture asserts that \(|\rho(G)| \le 3 \sigma(G)\), and that \(|\rho(G)| \le 2 \sigma(G)\) if \(G\) is solvable. Here the notation is as follows: for an irreducible character \(\chi\) of \(G\), \(\pi(\chi)\) denotes the set of prime divisors of \(\chi(1)\). Moreover, \(\rho(G) = \bigcup_{\chi \in \mathrm{Irr}(G)} \pi(\chi)\) and \(\sigma(G) = \max \{|\pi(\chi)|: \chi \in \mathrm{Irr}(G) \}\). The main result of the paper shows that \(|\rho(G)| \le 3 \sigma(G)\) if \(G\) is solvable. This improves the earlier bound \(|\rho(G)| \le 3 \sigma(G) + 2\) found by \textit{O. Manz} and \textit{T. R. Wolf} [Ill. J. Math. 37, No. 4, 652--665 (1993; Zbl 0832.20005)]. Reviewer's remark: The same bound was recently obtained by \textit{Z. Akhlaghi} et al. [J. Algebra 586, 537--560 (2021; Zbl 07385173)]. That paper also contains results on the non-solvable case of Huppert's conjecture.
scientific article; zbMATH DE number 7493673