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**On weakly \(s\)-permutably embedded subgroups of finite groups.**
*(English)*
Zbl 1177.20036

All groups considered are finite. Many authors have investigated the influence of some subgroups with certain embedding properties close to normality on the structure of the group. The paper under review is another contribution in this context. The authors introduce the following concept: a subgroup \(H\) of a group \(G\) is called weakly \(s\)-permutably embedded in \(G\) if there is a subnormal subgroup \(T\) of \(G\) and an \(s\)-permutably embedded subgroup \(H_{se}\) of \(G\) contained in \(H\) such that \(G=HT\) and \(H\cap T\leq H_{se}\). Following A. Ballester-Bolinches and M. C. Pedraza-Aguilera [in J. Pure Appl. Algebra 127, No. 2, 113-118 (1998; Zbl 0928.20020)], a subgroup \(H\) of \(G\) is said to be \(s\)-permutably embedded in \(G\) if every Sylow subgroup of \(H\) is a Sylow subgroup of some \(s\)-permutable (i.e., permutable with all Sylow subgroups) subgroup of \(G\). Clearly every \(s\)-permutably embedded subgroup is weakly \(s\)-permutably embedded. Moreover, weakly \(s\)-permutably embedding property also covers properly weakly \(s\)-permutability introduced by A. N. Skiba [in J. Algebra 315, No. 1, 192-209 (2007; Zbl 1130.20019)]: A subgroup \(H\) of a group \(G\) is weakly \(s\)-permutable in \(G\) if there is a subnormal subgroup \(T\) of \(G\) such that \(G=HT\) and \(H\cap T\leq H_{sG}\), where \(H_{sG}\) is the largest \(s\)-permutable subgroup of \(G\) contained in \(H\).

For a \(p\)-subgroup \(P\) of \(G\), \(p\) a prime number, the authors say that \(P\) satisfies \(\Delta_1\) in \(G\) if: \(P\) has a subgroup \(D\) such that \(1<|D|<|P|\) and all subgroups \(H\) of \(P\) with \(|H|=|D|\) are weakly \(s\)-permutably embedded in \(G\). When \(p=2\) and \(|P:D|>2\), in addition, \(H\) is weakly \(s\)-permutably embedded in \(G\) if there exists \(D_1\triangleleft H\leq P\) with \(2|D_1|=|D|\) and \(H/D_1\) is cyclic of order 4.

The main results in the paper are the following: Theorem 3.2. Let \(G\) be a group and \(P\) a Sylow \(p\)-subgroup of \(G\), where \(p\) is the smallest prime dividing \(|G|\). If \(P\) satisfies \(\Delta_1\) in \(G\), then \(G\) is \(p\)-nilpotent.

Theorem 3.5. Let \(\mathcal F\) be a saturated formation containing \(\mathcal U\), the class of all supersoluble groups, and \(G\) a group with a normal subgroup \(E\) such that \(G/E\in\mathcal F\). If every noncyclic Sylow subgroup of \(F^*(E)\) (the generalized Fitting subgroup of \(E\)) satisfies \(\Delta_1\) in \(G\), then \(G\in\mathcal F\).

A number of published results appear then as corollaries of these theorems.

For a \(p\)-subgroup \(P\) of \(G\), \(p\) a prime number, the authors say that \(P\) satisfies \(\Delta_1\) in \(G\) if: \(P\) has a subgroup \(D\) such that \(1<|D|<|P|\) and all subgroups \(H\) of \(P\) with \(|H|=|D|\) are weakly \(s\)-permutably embedded in \(G\). When \(p=2\) and \(|P:D|>2\), in addition, \(H\) is weakly \(s\)-permutably embedded in \(G\) if there exists \(D_1\triangleleft H\leq P\) with \(2|D_1|=|D|\) and \(H/D_1\) is cyclic of order 4.

The main results in the paper are the following: Theorem 3.2. Let \(G\) be a group and \(P\) a Sylow \(p\)-subgroup of \(G\), where \(p\) is the smallest prime dividing \(|G|\). If \(P\) satisfies \(\Delta_1\) in \(G\), then \(G\) is \(p\)-nilpotent.

Theorem 3.5. Let \(\mathcal F\) be a saturated formation containing \(\mathcal U\), the class of all supersoluble groups, and \(G\) a group with a normal subgroup \(E\) such that \(G/E\in\mathcal F\). If every noncyclic Sylow subgroup of \(F^*(E)\) (the generalized Fitting subgroup of \(E\)) satisfies \(\Delta_1\) in \(G\), then \(G\in\mathcal F\).

A number of published results appear then as corollaries of these theorems.

Reviewer: Ana Martínez-Pastor (Valencia)

### MSC:

20D40 | Products of subgroups of abstract finite groups |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |

20D35 | Subnormal subgroups of abstract finite groups |

20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |

20D15 | Finite nilpotent groups, \(p\)-groups |

### Keywords:

finite groups; saturated formations; \(p\)-nilpotent groups; weakly \(s\)-permutably embedded subgroups; subnormal subgroups; Sylow subgroups
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\textit{Y. Li} et al., Commun. Algebra 37, No. 3, 1086--1097 (2009; Zbl 1177.20036)

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