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Influence of normality on maximal subgroups of Sylow subgroups of a finite group. (English) Zbl 0802.20019
This paper deals with the influence of the normality of the maximal subgroups of Sylow \(p\)-subgroups of a finite group \(G\) on the structure of \(G\).
Here is a sample of the results obtained: 1) If \(G\) is a finite solvable group and every maximal subgroup of the Sylow subgroups of the Fitting subgroup \(F(G)\) is normal in \(G\), then \(G\) is supersolvable. 2) Let \(G\) be a finite group, let \(H \triangleleft G\) and assume that \(G/H\) is supersolvable and all maximal subgroups of the Sylow subgroups of \(H\) are normal in \(G\). Then \(G\) is supersolvable. 3) Let \(G\) be a finite group, let \(p = \max(\pi(G))\) and assume that every maximal subgroup of the Sylow \(q\)-subgroups of \(G\) is normal in \(G\) for all \(q \in \pi(G) - \{p\}\). Then \(G\) has a Sylow tower and \(G/O_ p(G)\) is supersolvable. In particular, \(G\) is solvable.

MSC:
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20E28 Maximal subgroups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D25 Special subgroups (Frattini, Fitting, etc.)
20D30 Series and lattices of subgroups
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References:
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