## Cover-avoidance properties and the structure of finite groups.(English)Zbl 1028.20014

Let $$G$$ be a finite group. The authors call a subgroup $$A$$ of the group $$G$$ a CAP-subgroup of $$G$$ if for any chief factor $$H/K$$ of $$G$$ $$H\cap A=K\cap A$$ or $$HA=KA$$. Some characterizations for a finite group to be solvable are obtained when some of its maximal subgroups or $$2$$-maximal subgroups are CAP-subgroups. In particular, it is proved that a group $$G$$ is solvable if and only if there exists a solvable $$2$$-maximal subgroup $$L$$ of $$G$$ such that $$L$$ is a CAP-subgroup of $$G$$.
It is also proved that $$G$$ is $$p$$-solvable if and only if a Sylow $$p$$-subgroup $$P$$ of $$G$$ is a CAP-subgroup of $$G$$.

### MSC:

 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D30 Series and lattices of subgroups 20E28 Maximal subgroups
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### References:

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