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On weakly \(s\)-permutable subgroups of finite groups. (English) Zbl 1130.20019
A subgroup \(H\) in a finite group \(G\) is weakly \(s\)-permutable in \(G\) if there exists a subnormal subgroup \(T\) such that \(G=HT\) and \(T\cap H\subseteq H_{sG}\), \(H_{sG}\) the subgroup of \(H\) generated by all subgroups of \(H\) that are \(s\)-permutable in \(G\). Consequently those classification results which require a subgroup to be either \(c\)-normal or \(s\)-permutable may be nontrivially generalized on the basis of being weakly \(s\)-permutable.
The author focuses on the prominent attributes of being weakly \(s\)-permutable underlying the hypotheses of twenty-four published results which become corollaries to the two principal theorems, a remarkable achievement that highlights the insight given to the development of the concept. The key is to fix in each noncyclic Sylow subgroup \(P\) of \(G\) a subgroup \(D\) that satisfies \(1<|D|<|P|\) and to study the structure of \(G\) under the assumption that each subgroup \(H\) with \(|H|=|D|\) is weakly \(s\)-permutable.

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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