On the modular version of Huppert's \(\rho\)-\(\sigma\) conjecture
Let \(p\) be a prime divisor of a finite solvable group \(G\), \(\text{IBr}_p(G)\) the set of all irreducible \(p\)-Brauer characters of \(G\). Let \(\sigma_p(G)=\max\{\sigma(\chi(1))\mid\chi\in\text{IBr}_p(G)\}\), where \(\sigma(n)\) is the number of distinct prime divisors of \(\chi(1)\). Next, \(\rho_p(G)\) denotes the set of prime divisors of irreducible \(p\)-Brauer characters of \(G\). There is the conjecture that \(|\rho_p(G)|\leq f(\sigma_p(G))\) for some function \(f\) (this is the \(\rho_p\)-\(\sigma_p\) conjecture). That conjecture for solvable groups is reduced to the case of nilpotent-by-metabelian groups. For the last groups, the authors find the function \(f\) in the following two cases: (1) the Sylow \(p\)-subgroups of \(G\) are cyclic. (2) \(\rho=2\) or \(p\) is a Fermat number.
scientific article; zbMATH DE number 1441804