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On \(c^*\)-normality and its properties. (English) Zbl 1125.20011
A subgroup \(H\) of a finite group \(G\) is said to be \(c^*\)-normal in \(G\) if there exists a subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K\) is S-quasinormally embedded in \(G\), that is, every Sylow subgroup of \(H\cap K\) is a Sylow subgroup of an S-quasinormal subgroup of \(G\).
The S-quasinormality was introduced and studied in 1998 by Pedraza-Aguilera and the reviewer and it has turned out useful to determine sufficient conditions on nilpotency, supersolubility and solubility. Similar role plays \(c\)-normality introduced by Wang in 1996. Since a \(c\)-normal subgroup \(H\) of a group \(G\) always has a supplement \(K\) such that \(H\cap K\) is just the core of \(H\) in \(G\), we may interpret \(c^*\)-normality as an extension of the \(c\)-normality. It is natural then to think about the influence of this subgroup embedding property on the structure of the group. This is the main goal of the present article, where sufficient conditions for \(p\)-nilpotency, \(p\)-supersolubility, supersolubility are proved. Extensions within the framework of formation theory are also presented.
The following result is typical: Theorem. Let \(F\) be a saturated formation containing the supersoluble groups. Suppose that \(G\) is a group with a normal subgroup \(H\) such that \(G/H\) belongs to \(F\). If all maximal subgroups of any Sylow subgroup of the generalised Fitting subgroup of \(H\) are \(c^*\)-normal in \(G\), then \(G\) belongs to \(F\).

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