## $$G$$-covering subgroup systems for the classes of supersoluble and nilpotent groups.(English)Zbl 1050.20009

Given a class of groups $$\mathfrak F$$, a set $$\Sigma$$ of subgroups of $$G$$ is called an $$\mathfrak F$$-covering subgroup system of $$G$$ if $$G\in{\mathfrak F}$$ whenever every subgroup in $$\Sigma$$ is in $$\mathfrak F$$. The authors are concerned with non-trivial sets of subgroups of a finite group $$G$$ which are $$\mathfrak F$$-covering subgroup systems for the classes $$\mathfrak F$$ of all supersoluble and all nilpotent subgroups.
In this paper, it is shown that given a soluble group $$G$$ with a normal subgroup $$N$$ such that $$G/N$$ is supersoluble, a set $$\Sigma$$ of subgroups of $$G$$ is a covering subgroup system for the class of all supersoluble groups if for every maximal subgroup $$M$$ of any Sylow subgroup of $$F(N)$$, either $$M$$ is normal in $$G$$ or $$\Sigma$$ contains a supplement of $$M$$ in $$G$$. A similar result is obtained when the condition is imposed on the maximal subgroups $$M$$ of the Sylow subgroups of $$N$$. From these results, some covering subgroup systems for the classes of all supersoluble and all nilpotent groups are derived. Finally, it is proved that if a set $$\Sigma$$ of subgroups of a group $$G$$ satisfies that for every cyclic subgroup $$L$$ of $$G$$ of prime order or order $$4$$ not contained in the Frattini subgroup of any other subgroup of $$G$$, either $$L$$ is normal in $$G$$ or $$\Sigma$$ contains a supplement of $$L$$ in $$G$$, then $$\Sigma$$ is a covering subgroup system for the class of supersoluble groups.

### MSC:

 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D15 Finite nilpotent groups, $$p$$-groups 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure

### Keywords:

finite groups; maximal subgroups; Sylow subgroups
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### References:

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