Finite groups with some maximal subgroups of Sylow subgroups \(\mathcal M\)-supplemented. (English. Russian original) Zbl 1184.20016
Math. Notes 86, No. 5, 655-664 (2009); translation from Mat. Zametki 86, No. 5, 692-704 (2009).
Summary: A subgroup \(H\) of a group \(G\) is said to be \(\mathcal M\)-supplemented in \(G\) if there exists a subgroup \(B\) of \(G\) such that \(G=HB\) and \(TB<G\) for every maximal subgroup \(T\) of \(H\). In this paper, we obtain the following statement: Let \(\mathcal F\) be a saturated formation containing all supersolvable groups and \(H\) be a normal subgroup of \(G\) such that \(G/H\in\mathcal F\). Suppose that every maximal subgroup of a noncyclic Sylow subgroup of \(F^*(H)\), having no supersolvable supplement in \(G\), is \(\mathcal M\)-supplemented in \(G\). Then \(G\in\mathcal F\).
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D40 | Products of subgroups of abstract finite groups |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
Keywords:
Sylow subgroups; supplemented subgroups; saturated formations; finite groups; supersolvable groups; Hall subgroups; Fitting subgroup; \(p\)-nilpotent groups; maximal subgroupsReferences:
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