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Nearly \(s\)-normal subgroups of a finite group. (English) Zbl 1170.20015
Summary: A subgroup \(H\) of a group \(G\) is said to be nearly \(s\)-normal in \(G\) if there exists a normal subgroup \(N\) of \(G\) such that \(HN\) is a normal subgroup of \(G\) and \(H\cap N\leq H_{sG}\), where \(H_{sG}\) is the largest \(s\)-permutable subgroup of \(G\) contained in \(H\). We obtain some results on nearly \(s\)-normal subgroups and supersoluble groups are hence characterized by this kind of subgroups.

MSC:
20D40 Products of 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
20D25 Special subgroups (Frattini, Fitting, etc.)
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