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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.

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.)