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On weakly \(s\)-permutably embedded subgroups. (English) Zbl 1222.20014

Summary: Let us suppose that \(G\) is a finite group and \(H\) is a subgroup of \(G\). \(H\) is said to be \(s\)-permutably embedded in \(G\) if for each prime \(p\) dividing \(|H|\), a Sylow \(p\)-subgroup of \(H\) is also a Sylow \(p\)-subgroup of some \(s\)-permutable subgroup of \(G\); \(H\) is called weakly \(s\)-permutably embedded in \(G\) if there are a subnormal subgroup \(T\) of \(G\) and an \(s\)-permutably embedded subgroup \(H_{se}\) of \(G\) contained in \(H\) such that \(G=HT\) and \(H\cap T\leq H_{se}\). We investigate the influence of weakly \(s\)-permutably embedded subgroups on the \(p\)-nilpotency and \(p\)-supersolvability of finite groups.

MSC:

20D40 Products of subgroups of abstract finite groups
20D35 Subnormal subgroups of abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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