## 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