On weakly \(S\)-embedded and weakly \(\tau\)-embedded subgroups.

*(English. Russian original)*Zbl 1311.20022
Sib. Math. J. 54, No. 5, 931-945 (2013); translation from Sib. Mat. Zh. 54, No. 5, 1162-1181 (2013).

In the paper under review, some subgroup embedding properties are studied in order to obtain information about groups in which some subgroups of prime power order satisfy these properties.

Let \(G\) be a finite group. A subgroup of \(G\) which permutes with all Sylow subgroups of \(G\) is called S-quasinormal (or S-permutable). A subgroup \(H\) of \(G\) is said to be S-quasinormally embedded (or S-permutably embedded) in \(G\) if every Sylow subgroup of \(H\) is a Sylow subgroup of some S-quasinormal subgroup of \(G\). A subgroup \(H\) of \(G\) is said to be weakly S-embedded in \(G\) if there exists a normal subgroup \(K\) such that \(HK\) is S-quasinormal in \(G\) and \(H\cap K\leq H_{seG}\), where \(H_{seG}\) is the subgroup generated by all subgroups of \(H\) which are S-quasinormally embedded in \(G\). A subgroup \(H\) of \(G\) is said to be \(\tau\)-quasinormal in \(G\) if \(H\) permutes with all Sylow \(q\)-subgroups \(Q\) of \(G\) such that \(\gcd(q,|H|)=1\) and \(\gcd(|H|,|Q^G|)\neq 1\). A subgroup \(H\) of a group \(G\) is said to be weakly \(\tau\)-embedded in \(G\) if there exists a normal subgroup \(K\) of \(G\) such that \(HK\) is S-quasinormal in \(G\) and \(H\cap K\leq H_{\tau G}\), where \(H_{\tau G}\) is the subgroup generated by all \(\tau\)-quasinormal subgroups of \(G\). The embedding properties analysed in this paper are weak S-embedding and weak \(\tau\)-embedding. We will denote by \(G^{\mathfrak N_p}\) the smallest normal subgroup \(N\) of \(G\) such that \(G/N\) is \(p\)-nilpotent.

The general hypothesis for Theorem 3.1 and Theorem 3.2 is that \(p\) is a prime divisor of \(|G|\) and that \(\gcd(|G|,(p-1)(p^2-1)\cdots (p^n-1))=1\) for some integer \(n\geq 1\). In Theorem 3.1, it is proved that \(G\) is \(p\)-nilpotent if and only if there exists a normal subgroup \(H\) of \(G\) such that \(G/H\) is \(p\)-nilpotent and for all Sylow \(p\)-subgroups \(P\) of \(H\), every \(n\)-maximal subgroup of \(P\) not containing \(P\cap G^{\mathfrak N_p}\) (if it exists) either has a \(p\)-nilpotent supplement in \(G\) or \(H\) is weakly S-embedded (and analogously with weakly \(\tau\)-embedded). A similar characterisation holds for subgroups \(L\) of order \(p^n\) or \(4\) (if \(p=2\) and \(n=1\), \(P\) is non-abelian and \(L\) is cyclic) not contained in \(Z_\infty(G)\) (Theorem 3.2). For \(A_4\)-free groups and primes \(p\) dividing \(|G|\) such that \(\gcd(|G|,p-1)=1\), a similar characterisation with \(2\)-maximal subgroups is obtained.

Recall that a saturated formation is a class of groups which is closed under epimorphic images, subdirect products, and Frattini extensions. For a saturated formation \(\mathfrak F\) containing all supersoluble groups, it is proved that \(G\in\mathfrak F\) if and only if there exists a normal subgroup \(H\) of \(G\) such that \(G/F\in\mathfrak F\) and, for any non-cyclic Sylow subgroup \(P\) of \(H\), each maximal subgroup of \(P\) either has a supersoluble supplement in \(G\) or is weakly S-embedded (or weakly \(\tau\)-embedded) in \(G\) (Theorem 3.4). A similar result is obtained with the condition that every cyclic subgroup \(L\) of \(P\) of prime order or order \(4\) (when \(p=2\) and \(P\) is non-abelian) not contained in \(Z_\infty(G)\) either has a supersoluble supplement in \(G\) or is weakly S-embedded (or weakly \(\tau\)-embedded), this is Theorem 3.5. Finally, similar results are obtained for Sylow subgroups of \(F^*(G)\), the generalised Fitting subgroup of \(G\) (Theorem 3.6).

Let \(G\) be a finite group. A subgroup of \(G\) which permutes with all Sylow subgroups of \(G\) is called S-quasinormal (or S-permutable). A subgroup \(H\) of \(G\) is said to be S-quasinormally embedded (or S-permutably embedded) in \(G\) if every Sylow subgroup of \(H\) is a Sylow subgroup of some S-quasinormal subgroup of \(G\). A subgroup \(H\) of \(G\) is said to be weakly S-embedded in \(G\) if there exists a normal subgroup \(K\) such that \(HK\) is S-quasinormal in \(G\) and \(H\cap K\leq H_{seG}\), where \(H_{seG}\) is the subgroup generated by all subgroups of \(H\) which are S-quasinormally embedded in \(G\). A subgroup \(H\) of \(G\) is said to be \(\tau\)-quasinormal in \(G\) if \(H\) permutes with all Sylow \(q\)-subgroups \(Q\) of \(G\) such that \(\gcd(q,|H|)=1\) and \(\gcd(|H|,|Q^G|)\neq 1\). A subgroup \(H\) of a group \(G\) is said to be weakly \(\tau\)-embedded in \(G\) if there exists a normal subgroup \(K\) of \(G\) such that \(HK\) is S-quasinormal in \(G\) and \(H\cap K\leq H_{\tau G}\), where \(H_{\tau G}\) is the subgroup generated by all \(\tau\)-quasinormal subgroups of \(G\). The embedding properties analysed in this paper are weak S-embedding and weak \(\tau\)-embedding. We will denote by \(G^{\mathfrak N_p}\) the smallest normal subgroup \(N\) of \(G\) such that \(G/N\) is \(p\)-nilpotent.

The general hypothesis for Theorem 3.1 and Theorem 3.2 is that \(p\) is a prime divisor of \(|G|\) and that \(\gcd(|G|,(p-1)(p^2-1)\cdots (p^n-1))=1\) for some integer \(n\geq 1\). In Theorem 3.1, it is proved that \(G\) is \(p\)-nilpotent if and only if there exists a normal subgroup \(H\) of \(G\) such that \(G/H\) is \(p\)-nilpotent and for all Sylow \(p\)-subgroups \(P\) of \(H\), every \(n\)-maximal subgroup of \(P\) not containing \(P\cap G^{\mathfrak N_p}\) (if it exists) either has a \(p\)-nilpotent supplement in \(G\) or \(H\) is weakly S-embedded (and analogously with weakly \(\tau\)-embedded). A similar characterisation holds for subgroups \(L\) of order \(p^n\) or \(4\) (if \(p=2\) and \(n=1\), \(P\) is non-abelian and \(L\) is cyclic) not contained in \(Z_\infty(G)\) (Theorem 3.2). For \(A_4\)-free groups and primes \(p\) dividing \(|G|\) such that \(\gcd(|G|,p-1)=1\), a similar characterisation with \(2\)-maximal subgroups is obtained.

Recall that a saturated formation is a class of groups which is closed under epimorphic images, subdirect products, and Frattini extensions. For a saturated formation \(\mathfrak F\) containing all supersoluble groups, it is proved that \(G\in\mathfrak F\) if and only if there exists a normal subgroup \(H\) of \(G\) such that \(G/F\in\mathfrak F\) and, for any non-cyclic Sylow subgroup \(P\) of \(H\), each maximal subgroup of \(P\) either has a supersoluble supplement in \(G\) or is weakly S-embedded (or weakly \(\tau\)-embedded) in \(G\) (Theorem 3.4). A similar result is obtained with the condition that every cyclic subgroup \(L\) of \(P\) of prime order or order \(4\) (when \(p=2\) and \(P\) is non-abelian) not contained in \(Z_\infty(G)\) either has a supersoluble supplement in \(G\) or is weakly S-embedded (or weakly \(\tau\)-embedded), this is Theorem 3.5. Finally, similar results are obtained for Sylow subgroups of \(F^*(G)\), the generalised Fitting subgroup of \(G\) (Theorem 3.6).

Reviewer: Ramón Esteban-Romero (València)

##### MSC:

20D40 | Products of subgroups of abstract finite groups |

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

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

20D25 | Special subgroups (Frattini, Fitting, etc.) |

##### Keywords:

finite groups; subgroup embedding properties; weakly \(S\)-embedded subgroups; weakly \(\tau\)-embedded subgroups; saturated formations; Sylow subgroups; quasinormal subgroups; permutable subgroups; \(p\)-nilpotent supplements##### References:

[1] | Huppert B., Endliche Gruppen. I, Springer-Verlag, Berlin, Heidelberg, and New York (1979). |

[2] | Robinson D. J. S., A Course in the Theory of Groups, Springer-Verlag, New York (1996). |

[3] | Guo W., The Theory of Classes of Groups, Science Press-Kluwer Acad. Publ., Beijing, New York, Dordrecht, Boston, and London (2000). |

[4] | Kegel O. H., ”Sylow-Gruppen and Subnormalteiler endlicher Gruppen,” Math. Z., 78, 205–221 (1962). · Zbl 0102.26802 |

[5] | Ballester-Bolinches A. and Pedraza-Aguilera M. C., ”Sufficient conditions for supersolubility of finite groups,” J. Pure Appl. Algebra, 127, 113–118 (1998). · Zbl 0928.20020 |

[6] | Li S., Shen Z., Liu J., and Liu X., ”The influence of SS-quasinormality of some subgroups on the structure of finite groups.,” J. Algebra, 319, 4275–4287 (2008). · Zbl 1152.20019 |

[7] | Chen Z., ”On a theorem of Srinivasan,” J. Southwest Normal Univ. Nat. Sci., 2, No. 1, 1–4 (1987). · Zbl 0732.20008 |

[8] | Lukyanenko V. O. and Skiba A. N., ”On {\(\tau\)}-quasinormal and weakly {\(\tau\)}-quasinormal subgroups of finite groups,” Math. Sc. Res. J., 2, 243–257 (2008). · Zbl 1179.20018 |

[9] | Guo W., Shum K. P., and Skiba A. N., ”On solubility and supersolubility of some classes of finite groups.,” Sci. China Ser. A: Math., 52, No. 1, 1–15 (2009). · Zbl 1189.20023 |

[10] | Li J., Chen G., and Chen R., ”On weakly S-embedded subgroups of finite groups,” Sci. China Math., 54, No. 9, 1899–1908 (2011). · Zbl 1239.20025 |

[11] | Deskins W. E., ”On quasinormal subgroups of finite groups,” Math. Z., 82, No. 2, 125–132 (1963). · Zbl 0114.02004 |

[12] | Schmid P., ”Subgroups permutable with all Sylow subgroups,” J. Algebra, 82, No. 1 1, 285–293 (1998). · Zbl 0910.20015 |

[13] | Lukyanenko V. O. and Skiba A. N., ”On weakly {\(\tau\)}-quasinormal subgroups of finite groups,” Acta Math. Hungar., 25, No. 3, 247–248 (2009). · Zbl 1207.20009 |

[14] | Li Y., Wang Y., and Wei H., ”On p-nilpotency of finite groups with some subgroups {\(\pi\)}-quasinormally embedded,” Acta Math. Hungar., 08, No. 4, 283–298 (2005). · Zbl 1094.20007 |

[15] | Doerk K. and Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin and New York (1992). · Zbl 0753.20001 |

[16] | Gross F., ”Conjugacy of odd order Hall subgroups,” Bull. London Math. Soc., 9, No. 4, 311–319 (1987). · Zbl 0616.20007 |

[17] | Wang Y., Wei H., and Li Y., ”A generalization of Kramer’s theorem and its applications,” Bull. Austral. Math. Soc., 65, 467–475 (2002). · Zbl 1012.20010 |

[18] | Tate J., ”Nilpotent quotient groups,” Topology, 3, 109–111 (1964). · Zbl 0125.01503 |

[19] | Guo W., ”On F-supplemented subgroups of finite groups,” Manuscripta Math., 27, 139–150 (2008). · Zbl 1172.20019 |

[20] | Guo W. and Skiba A. N., ”Finite groups with given s-embedded and n-embedded subgroups,” J. Algebra, 321, 2843–2860 (2009). · Zbl 1182.20026 |

[21] | Huppert B. and Blackburn N., Finite Groups. III, Springer-Verlag, Berlin (1982). · Zbl 0514.20002 |

[22] | Wang Y., ”On c-normality of groups and its properties,” J. Algebra, 180, No. 3, 954–965 (1996). · Zbl 0847.20010 |

[23] | Wei H. and Wang Y., ”On c*-normality and its properties,” J. Group Theory, 10, 211–224 (2007). · Zbl 1125.20011 |

[24] | Wang L. and Wang Y., ”On S-semipermutable maximal and minimal subgroups of Sylow p-subgroups of finite groups.,” Comm. Algebra, 34, 143–149 (2006). · Zbl 1087.20015 |

[25] | Guo W., Lu Y., and Niu W., ”S-embedded subgroups of finite groups,” Algebra and Logic, 49, No. 4, 293–304 (2010). · Zbl 1255.20021 |

[26] | Zhang Q. and Wang L., ”The influence of S-semipermutability of some subgroups on the structure of groups,” Acta Math. Sin., 48, No. 1, 81–88 (2005). · Zbl 1119.20026 |

[27] | Li S., Shen Z., and Kong X., ”On SS-quasinormal subgroups of finite groups,” Comm. Algebra, 36, 4436–4447 (2008). · Zbl 1163.20011 |

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