Finite groups in which all \(p\)-subnormal subgroups form a lattice. (English) Zbl 0866.20013
O. Kegel, in 1962, introduced the concept of \(p\)-subnormal subgroups of a finite group as the subgroups whose intersections with all Sylow \(p\)-subgroups of the group are Sylow \(p\)-subgroups of the subgroup. The set of \(p\)-subnormal subgroups of a finite group is not a lattice in general. In this paper, the class of all finite groups in which all \(p\)-subnormal subgroups form a lattice is determined. This is the class of all finite \(p\)-soluble groups whose \(p\)-length and \(p'\)-length, both, are less or equal to 1. The join-semilattice case and the meet-semilattice case are analyzed separately.
Reviewer: L.M.Ezquerro (Pamplona)
MSC:
| 20D30 | Series and lattices of subgroups |
| 20D35 | Subnormal subgroups of abstract finite groups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
Keywords:
\(p\)-subnormal subgroups; Sylow \(p\)-subgroups; finite \(p\)-soluble groups; \(p\)-length; join-semilattices; meet-semilatticesReferences:
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