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On $$c$$-normal maximal and minimal subgroups of Sylow subgroups of finite groups. II. (English) Zbl 1050.20011
[For part I cf. the first author, Commun. Algebra 29, No. 5, 2193-2200 (2001; Zbl 0990.20012).]
A subgroup $$H$$ of a finite group $$G$$ is said to be $$c$$-normal in $$G$$ if there exists a subgroup $$N$$ of $$G$$ such that $$G=HN$$ and $$H\cap N$$ is contained in $$\text{Core}_G(H)$$, the largest normal subgroup of $$G$$ contained in $$H$$. Given a group $$G$$, the authors study the influence of the $$c$$-normality of the maximal and the minimal subgroups of the Sylow subgroups of the generalised Fitting subgroup of some normal subgroup of $$G$$ on the structure of $$G$$. For a saturated formation $$\mathfrak F$$ containing the class of all supersoluble groups, they prove that if $$H$$ is a normal subgroup of a group $$G$$ such that $$G/H\in{\mathfrak F}$$ and all maximal subgroups of all Sylow subgroups of the generalised Fitting subgroup $$F^*(H)$$ of $$H$$ are $$c$$-normal in $$G$$, then $$G\in{\mathfrak F}$$, and that if all subgroups of order a prime number or $$4$$ of $$F^*(H)$$ are $$c$$-normal in $$G$$, then $$G\in{\mathfrak F}$$.
Reviewer’s remark: There are some misprints in the paper, e.g., in the statement of Theorem 3.2, where the $$c$$-normality is imposed on all minimal subgroups and all cyclic subgroups of $$F^*(H)$$, but it is actually needed only for all minimal subgroups and all cyclic subgroups of order $$4$$ of $$F^*(H)$$.

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