A dual version of Huppert’s \(\rho\)-\(\sigma\) conjecture. (English) Zbl 1133.20005
The main result of the paper is the following: There exists an integer-valued function \(f\) such that if \(G\) is a finite group such that for every prime \(p\) there are at most \(k\) members of \(\text{cd}(G)\) which are divisible by \(p\), then \(|\text{cd}(G)|\leq f(k)\). Here \(\text{cd}(G)\) denotes the set of the irreducible complex character degrees of \(G\). The proof depends on CFSG.
An explicit function \(f\) could be obtained from the paper, but, according to the authors, would be far from best possible. This paper extends earlier work by D. Benjamin and J. McVey on the problem for solvable groups, for which a quadratic bound had been obtained [see D. Benjamin, Proc. Am. Math. Soc. 125, No. 10, 2831-2837 (1997; Zbl 0889.20004)].
An explicit function \(f\) could be obtained from the paper, but, according to the authors, would be far from best possible. This paper extends earlier work by D. Benjamin and J. McVey on the problem for solvable groups, for which a quadratic bound had been obtained [see D. Benjamin, Proc. Am. Math. Soc. 125, No. 10, 2831-2837 (1997; Zbl 0889.20004)].
Reviewer: Thomas Michael Keller (San Marcos)
MSC:
| 20C15 | Ordinary representations and characters |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
