Big primes and character values for solvable groups
It was shown by \textit{F. M. Markel} [ibid. 26, 69-74 (1973; Zbl 0274.20030)], that if the finite supersolvable group G has all its character values in \({\mathbb{Q}}\), then no prime bigger than 3 divides the order of G. This was extended by \textit{R. Gow} [ibid. 40, 280-299 (1976; Zbl 0348.20017)], who proved that for a finite solvable group G having all its character values in \({\mathbb{Q}}\), the order of G is not divisible by any prime bigger than 5. This work extends the above results in several directions. One of the main results is Theorem: Let G be a finite solvable group whose character values lie all in a field of degree n over \({\mathbb{Q}}\) and let p be a prime divisor of the order of G. Then \(p\leq 16n^ 2+1\). In fact, there are several theorems proved here in the same spirit.
scientific article; zbMATH DE number 3941772