On the maximal overgroups of Sylow subgroups of finite groups.(English)Zbl 1537.20032
Let \(G\) be a finite group, \(r \in \pi(G)\), \(R\) a Sylow \(r\)-subgroup and let \(\mathcal{M}(R)\) be the set of maximal subgroups of \(G\) containing \(R\). The main goal of the paper under review is to determine the triples \((G,r,H)\) with \(\mathcal{M}(R)=\{H\}\). The proof involves a reduction to almost simple groups and the main theorem (which cannot be stated here) extends earlier work ofM. Aschbacher [J. Algebra 66, 400–424 (1980;Zbl 0445.20008)] in the special case \(r=2\).
The authors also present several applications. In particular, they prove some new results on weakly subnormal subgroups of finite groups, which can be used to study variations of the Baer-Suzuki theorem.
The authors also present several applications. In particular, they prove some new results on weakly subnormal subgroups of finite groups, which can be used to study variations of the Baer-Suzuki theorem.
Reviewer: Enrico Jabara (Venezia)
MSC:
| 20D05 | Finite simple groups and their classification |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D30 | Series and lattices of subgroups |
| 20E28 | Maximal subgroups |
| 20B15 | Primitive groups |
| 20G40 | Linear algebraic groups over finite fields |
Citations:
Zbl 0445.20008Cite
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