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$$SE$$-supplemented subgroups of finite groups. (English) Zbl 1287.20025
Let $$G$$ be a finite group, $$H$$ its subgroup, $$\text{Syl}(G)$$ the set of all Sylow-subgroups. $$H$$ is called S-quasinormal (S-quasinormally embedded) if $$HP=PH$$ for all $$P\in\text{Syl}(G)$$ (if every $$P\in\text{Syl}(H)$$ is in $$\text{Syl}(S)$$ for some S-quasinormal subgroup $$S\leq G$$). The subgroup $$H_s=\langle\bigcup\{S\leq H\mid S\text{ is S-quasinormally embedded in }G\}\rangle$$ is called the SE-core of $$H\leq G$$. $$H$$ is called SE-supplemented if for some $$T\leq G$$, $$HT=G$$ and $$H\cap T\leq H_s$$.
The question whether given a saturated formation $$\mathcal F$$ containing all supersolvable groups, if for some normal subgroup $$E\leq G$$, $$G/E\in\mathcal F$$, under what conditions on $$E$$, $$G\in\mathcal F$$, has been investigated recently by many authors such as A. Ballester-Bolinches, Y. Wang and X. Guo [Glasg. Math. J. 42, No. 3, 383-389 (2000; Zbl 0968.20009)] and H. Wei, Y. Wang and Y. Li [Proc. Am. Math. Soc. 132, No. 8, 2197-2204 (2004; Zbl 1061.20014)].
The authors extending the results of these and other papers, answer the following questions affirmatively: Is it true that the already known results can be extended to Baer-local formations or strengthened by weakening conditions? Namely, the two main results are as follows: given a Baer-local formation $$\mathcal F$$ containing all supersolvable groups and $$E\leq G$$ a normal subgroup with $$G/E\in\mathcal F$$, if all cyclic subgroups of prime order or of order 4 (all maximal subgroups of all $$P\in\text{Syl}(E)$$) are SE-supplemented then $$G\in\mathcal F$$.

##### MSC:
 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D40 Products of subgroups of abstract finite groups 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D35 Subnormal subgroups of abstract finite groups
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##### References:
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