Finite groups in which all p-subnormal subgroups form a lattice
Copyright policy )Finite groups in which all p-subnormal subgroups form a lattice
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.
Sylow \(p\)-subgroups, join-semilattices, Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure, finite \(p\)-soluble groups, Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks, \(p\)-subnormal subgroups, Series and lattices of subgroups, \(p\)-length, Subnormal subgroups of abstract finite groups, meet-semilattices
Sylow \(p\)-subgroups, join-semilattices, Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure, finite \(p\)-soluble groups, Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks, \(p\)-subnormal subgroups, Series and lattices of subgroups, \(p\)-length, Subnormal subgroups of abstract finite groups, meet-semilattices
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