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On \(NR^*\)-subgroups of finite groups. (English) Zbl 1292.20018

Summary: Let \(G\) be a finite group and let \(H\) be a subgroup of \(G\). \(H\) is said to be an \(NR^*\)-subgroup of \(G\) if there exists a normal subgroup \(T\) of G such that \(G=HT\) and if whenever \(K\triangleleft H\) and \(g\in G\), then \(K^g\cap H\cap T\leq K\). A number of new characterizations of a group \(G\) are given, under the assumption that all Sylow subgroups of certain subgroups of \(G\) are \(NR^*\)-subgroups.

MSC:

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.)
20D40 Products of subgroups of abstract finite groups
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