Finite soluble groups with supersoluble Sylow normalizers. (English) Zbl 0638.20013
Let G be a finite group. It was proved by M. Bianchi, A. Gillio Berta Mauri and 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 \(| \pi (G)| \geq 3:\) \(l_ p(G)\leq 2\).
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 \(| \pi (G)| \geq 3:\) \(l_ p(G)\leq 2\).
Reviewer: M.Deaconescu
MSC:
20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
20D25 | Special subgroups (Frattini, Fitting, etc.) |
Citations:
Zbl 0605.20017References:
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