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On \(sse\)-embedded subgroups of finite groups. (English) Zbl 1442.20011
In recent year, the normality of subgroup of a finite group is extended continuously in the literature. In this paper, the generalized normal subgroup is called an \(sse\)-embedded subgroup. Let \(G\) be a finite group and \(H\) be a subgroup of \(G\). Then \(H\) is said to be \(sse\)-embedded in \(G\) if there exists a subgroup \(T\) in \(G\) such that \(HT\) is an \(s\)-permutable subgroup of \(G\) and \(H \cap T\leq H_{ssG}\), where \(H_{ssG}\) is an \(s\)-semipermutable subgroup of \(G\) contained in \(H\). By assuming some class of subgroups of a finite group \(G\) to be \(sse\)-embedded subgroups, some sufficient conditions for the solubility of \(G\) are given.
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
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