## On $$c$$-normality of certain subgroups of prime power order of finite groups.(English)Zbl 1082.20008

Let $$G$$ be a finite group. A subgroup $$H$$ of $$G$$ is called $$c$$-normal in $$G$$ if $$G$$ has a normal subgroup $$N$$ such that $$G=NH$$ and $$N\cap H\leq\text{Core}(H)$$, where $$\text{Core}(H)$$ is the intersection of all the conjugates of $$H$$ in $$G$$.
The authors prove the following main results: Let $$\mathcal F$$ be a saturated formation containing the formation of supersoluble groups. Then $$G\in\mathcal F$$ if and only if there is a normal subgroup $$H$$ of $$G$$ such that $$G/H\in\mathcal F$$ and all maximal subgroups of the Sylow subgroups of $$H$$ are $$c$$-normal in $$G$$ (Theorem 3.3). The same result holds if we replace the maximal subgroups of the Sylow subgroups of $$H$$ by the subgroups of prime order or order $$4$$ of $$H$$ (Theorem 3.9).
The proofs of the above results rely on the following $$p$$-supersolubility criterion: Let $$G$$ be a finite $$p$$-soluble group for a prime $$p$$. Then $$G$$ is $$p$$-supersoluble if there is a normal subgroup $$H$$ of $$G$$ such that $$G/H$$ is $$p$$-supersoluble and either all maximal subgroups of the Sylow subgroups of $$H$$ are $$c$$-normal in $$G$$ or the subgroups of prime order or order $$4$$ of $$H$$ are $$c$$-normal in $$G$$ (Theorems 3.1 and 3.7).

### MSC:

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