Finite soluble groups with supersoluble Sylow normalizers
Let G be a finite group. It was proved by \textit{M. Bianchi}, \textit{A. Gillio Berta Mauri} and \textit{P. Hauck} [Arch. Math. 47, 193-197 (1986; Zbl 0605.20017)] that if every normalizer of Sylow p-subgroups of G (p\(\in \pi (G))\) is nilpotent, then G is nilpotent. The corresponding result is false if one relaxes the condition of nilpotency to that of supersolvability, since there are simple nonabelian groups having supersolvable Sylow normalizers. The authors consider the class \(N^{{\mathcal S}}\) of the finite groups all of whose Sylow normalizers are supersolvable and restrict to the solvable members of \(N^{{\mathcal L}}\). If \(G\in N^{{\mathcal S}}\) and \(\pi (G)=\{p,q\}\), \(p| q-1\), they prove that there exists \(H\in N^{{\mathcal S}}\), with \(\pi (H)=\pi (G)\) and \(X\leq H\) with \(X\cong G\). If G is soluble, \(G\in N^{{\mathcal S}}\) and \(| \pi (G)| \geq 3\), then it is not always possible to embed G in this manner in a solvable group \(H\in N^{{\mathcal S}}\) with \(\pi (H)=\pi (G)\). The following bound for the Fitting length h(G) is obtained for a solvable group \(G\in N^{{\mathcal S}}\); if \(\pi (G)=\{p_ 1,p_ 2,...,p_ n\}\), if \(l_ i(G)\leq l_ j(G)\) for \(i<j\), where \(l_ i(G)\) is the \(p_ i\)-length of G, then \(h(G)\leq l_ 1(G)+l_ 2(G)\). The smallest member p of \(\pi\) (G) has a special property if \(G\in N^{{\mathcal S}}\) is solvable and \(| \pi (G)| \geq 3:\) \(l_ p(G)\leq 2\).
scientific article; zbMATH DE number 4040162
scientific article