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On $$\mathcal U_c$$-normal subgroups of finite groups. (English) Zbl 1126.20012
The authors deal in the paper with finite groups and introduce the following concept: Let $$G$$ be a group. A subgroup $$H$$ of $$G$$ is $$\mathcal U_{\text{c}}$$-normal in $$G$$ if $$G$$ has a subnormal subgroup $$T$$ such that $$TH=G$$ and $$(H\cap T)H_G/H_G$$ is contained in the $$\mathcal U$$-hypercenter $$Z_\infty^{\mathcal U}(G/H_G)$$ of $$G/H_G$$, where $$\mathcal U$$ is the class of the finite supersoluble groups. This subgroup is the largest normal subgroup of $$G$$ whose $$G$$-chief factors are cyclic.
The paper contains interesting results about the influence of this subgroup embedding property on the stucture of the groups. The following two results are the main ones:
Theorem. Let $$\mathcal F$$ be a saturated formation containing the supersoluble groups. If $$G$$ is a group with a normal subgroup $$E$$ such that $$G/E$$ is an $$\mathcal F$$-group and every maximal subgroup of every non-cyclic Sylow subgroup of $$E$$ not having a supersoluble supplement in $$G$$ is $$\mathcal U_{\text{c}}$$-normal in $$G$$, then $$G$$ belongs to $$\mathcal F$$.
Theorem. Let $$\mathcal F$$ be a subgroup-closed saturated formation containing the supersoluble groups. If $$G$$ is a group with a normal subgroup $$E$$ such that $$G/E$$ is an $$\mathcal F$$-group and all cyclic subgroups of $$E$$ of prime or order $$4$$ are $$\mathcal U_{\text{c}}$$-normal in $$G$$, then $$G$$ belongs to $$\mathcal F$$.

##### 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 20D40 Products of subgroups of abstract finite groups
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