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Finite groups with given c-permutable subgroups. (English) Zbl 1067.20018
Summary: We say that subgroups \(H\) and \(T\) of a group \(G\) are c-permutable in \(G\) if there exists an element \(x\in G\) such that \(HT^x=T^xH\). We prove that a finite soluble group \(G\) is supersoluble if and only if every maximal subgroup of every Sylow subgroup of \(G\) is c-permutable with all Hall subgroups of \(G\).

MSC:
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D40 Products of subgroups of abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20E28 Maximal subgroups
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