\(SE\)-supplemented subgroups of finite groups.

*(English)*Zbl 1287.20025Let \(G\) 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 \(G\in\mathcal F\).

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 \(G\in\mathcal F\).

Reviewer: János Kurdics (Nyíregyháza)

##### 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 |

##### Keywords:

finite groups; saturated formations; Baer-local formations; quasinormal subgroups; supplemented subgroups; Sylow subgroups; quasinormally embedded subgroups
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\textit{W. Guo} et al., Rend. Semin. Mat. Univ. Padova 129, 245--263 (2013; Zbl 1287.20025)

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