Charaktergrade und die Kommutatorlänge in auflösbaren Gruppen. (Character degrees and the derived length in soluble groups)
For \(n\in {\mathbb{N}}\) let \(\omega\) (n) denote the number of prime factors in the prime number decomposition of n counting multiplicities, and for a finite group G let \(\omega (G)=\max \{\omega (\chi (1))|\) \(\chi \in Irr_{{\mathbb{C}}}(G)\}\). Obviously, \(\omega (G)=0\) characterizes abelian groups. If \(\omega (G)=1\), then by results of Isaacs and Passman, G is solvable with derived length dl(G) bounded by 3. For G solvable with \(\omega\) (G)\(\geq 2\), \textit{B. Huppert} proved dl(G)\(\leq 2\omega (G)\) [Arch. Math. 46, 387-392 (1986; Zbl 0608.20003)]. The main result of the paper under review is an improvement of Huppert's inequality to \(dl(G)\leq [(3/2)\omega (G)+2].\) Better bounds are available for \(\omega (G)=3\) or 4, furthermore for groups of odd order.
scientific article; zbMATH DE number 4118577
scientific article