A note on \(S\)-semipermutable and \(S\)-permutably embedded subgroups of finite groups. (English) Zbl 07909930
Summary: In this note, we obtain some criteria for \(p\)-supersolvability and \(p\)-nilpotency of a finite group and extend some known results concerning \(S\)-semipermutable and \(S\)-permutably embedded subgroups. In particular, we generalize some main results of Shen et al. (J Group Theory 13(2):257-265, 2010) and Kong and Guo (Ric Mat 68(2):571-579, 2019).
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
Keywords:
\(p\)-supersolvable; \(p\)-nilpotent; formation residual; \(S\)-semipermutable subgroups; \(S\)-permutably embedded subgroupsReferences:
| [1] | Ballester-Bolinches, A.; Esteban-Romero, R.; Qiao, S., A note on a result of Guo and Isaacs about \(p\)-supersolubility of finite groups, Arch. Math. (Basel), 106, 6, 501-506, 2016 · Zbl 1342.20020 · doi:10.1007/s00013-016-0901-7 |
| [2] | Ballester-Bolinches, A.; Pedraza-Aguilera, MC, Sufficient conditions for supersolubility of finite groups, J. Pure Appl. Algebra, 127, 2, 113-118, 1998 · Zbl 0928.20020 · doi:10.1016/S0022-4049(96)00172-7 |
| [3] | Berkovich, Y.; Isaacs, IM, \(p\)-Supersolvability and actions on \(p\)-groups stabilizing certain subgroups, J. Algebra, 414, 82-94, 2014 · Zbl 1316.20012 · doi:10.1016/j.jalgebra.2014.04.026 |
| [4] | Chen, ZM, On a theorem of Srinivasan, J. Southwest Normal Univ. Nat. Sci., 12, 1, 1-4, 1987 · Zbl 0732.20008 |
| [5] | Guo, Y.; Isaacs, IM, Conditions on \(p\)-subgroups implying \(p\)-nilpotence or \(p\)-supersolvability, Arch. Math. (Basel), 105, 3, 215-222, 2015 · Zbl 1344.20032 · doi:10.1007/s00013-015-0803-0 |
| [6] | Huppert, B., Endliche Gruppen, 1967, Berlin: Springer, Berlin · Zbl 0217.07201 · doi:10.1007/978-3-642-64981-3 |
| [7] | Isaacs, IM, Finite Group Theory, Graduate Studies in Mathematics, 2008, Providence: American Mathematical Society, Providence · Zbl 1169.20001 |
| [8] | Isaacs, IM, Semipermutable \(\pi \)-subgroups, Arch. Math. (Basel), 102, 1, 1-6, 2014 · Zbl 1297.20018 · doi:10.1007/s00013-013-0604-2 |
| [9] | Kegel, OH, Sylow-Gruppen and Subnormalteiler endlicher Gruppen, Math. Z., 78, 205-221, 1962 · Zbl 0102.26802 · doi:10.1007/BF01195169 |
| [10] | Kong, Q.; Guo, X., On \(ss\)-quasinormal or weakly \(s\)-permutably embedded subgroups of finite groups, Monatsh. Math., 182, 3, 637-647, 2017 · Zbl 1365.20014 · doi:10.1007/s00605-016-0877-1 |
| [11] | Kong, Q.; Guo, X., On weakly \(s\)-semipermutable or \(ss\)-quasinormal subgroups of finite groups, Ric. Mat., 68, 2, 571-579, 2019 · Zbl 1453.20034 · doi:10.1007/s11587-018-0427-3 |
| [12] | Li, C.; Zhang, X., A note “On \(ss\)-quasinormal or weakly \(s\)-permutably embedded subgroups of finite groups”, Monatsh. Math., 183, 1, 159-163, 2017 · Zbl 1367.20017 · doi:10.1007/s00605-016-0913-1 |
| [13] | Li, S.; He, X., On normally embedded subgroups of prime power order in finite groups, Commun. Algebra, 36, 6, 2333-2340, 2008 · Zbl 1146.20015 · doi:10.1080/00927870701509370 |
| [14] | Li, S.; Shen, Z.; Liu, J.; 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 · doi:10.1016/j.jalgebra.2008.01.030 |
| [15] | Li, Y.; Qiao, S.; Su, N.; Wang, Y., On weakly \(s\)-semipermutable subgroups of finite groups, J. Algebra, 371, 250-261, 2012 · Zbl 1269.20020 · doi:10.1016/j.jalgebra.2012.06.025 |
| [16] | Li, Y.; Qiao, S.; Wang, Y., On weakly \(s\)-permutably embedded subgroups of finite groups, Commun. Algebra, 37, 3, 1086-1097, 2009 · Zbl 1177.20036 · doi:10.1080/00927870802231197 |
| [17] | Li, Y.; Wang, Y.; Wei, H., The influence of \(\pi \)-quasinormality of some subgroups of a finite group, Arch. Math. (Basel), 81, 3, 245-252, 2003 · Zbl 1053.20017 · doi:10.1007/s00013-003-0829-6 |
| [18] | Li, Y.; Wang, Y.; Wei, H., On \(p\)-nilpotency of finite groups with some subgroups \(\pi \)-quasinormally embedded, Acta Math. Hungar., 108, 4, 283-298, 2005 · Zbl 1094.20007 · doi:10.1007/s10474-005-0225-8 |
| [19] | Shen, Z.; Li, S.; Shi, W., Finite groups with normally embedded subgroups, J. Group Theory, 13, 2, 257-265, 2010 · Zbl 1196.20022 · doi:10.1515/jgt.2010.042 |
| [20] | Shen, Z.; Zhang, J.; Wu, S., Finite groups with weakly \(S\)-semipermutably embedded subgroups, Int. Electron. J. Algebra, 11, 111-124, 2012 · Zbl 1253.20015 |
| [21] | Shen, Z.; Zhang, J., Corrigendum to “Finite groups with weakly \(S\)-semipermutably embedded subgroups”, Int. Electron. J. Algebra, 12, 175-176, 2012 · Zbl 1253.20016 |
| [22] | Tang, N.; Li, X., On weakly \(s\)-semipermutable subgroups of finite groups, Algebra Colloq., 21, 4, 541-550, 2014 · Zbl 1309.20016 · doi:10.1142/S1005386714000492 |
| [23] | Wei, H.; Gu, W.; Pan, H., On \(c^{\ast }\)-normal subgroups in finite groups, Acta Math. Sin. (Engl. Ser.), 28, 3, 623-630, 2012 · Zbl 1254.20018 · doi:10.1007/s10114-012-9226-z |
| [24] | Wei, H.; Wang, Y., On \(c^{\ast }\)-normality and its properties, J. Group Theory, 10, 2, 211-223, 2007 · Zbl 1125.20011 · doi:10.1515/JGT.2007.017 |
| [25] | Wu, X.; Li, X., Weakly \(s\)-semipermutable subgroups and structure of finite groups, Commun. Algebra, 48, 6, 2307-2314, 2020 · Zbl 1484.20024 · doi:10.1080/00927872.2019.1711107 |
| [26] | Yu, H., On weakly \(S\)-permutably embedded subgroups of finite groups, Bull. Aust. Math. Soc., 94, 3, 437-448, 2016 · Zbl 1381.20019 · doi:10.1017/S0004972716000344 |
| [27] | Yu, H., Some sufficient and necessary conditions for \(p\)-supersolvablity and \(p\)-nilpotence of a finite group, J. Algebra Appl., 16, 3, 1750052, 2017 · Zbl 1375.20016 · doi:10.1142/S0219498817500529 |
| [28] | Yu, H., On generalized \(S\Phi \)-supplemented subgroups of finite groups, J. Algebra Appl., 18, 11, 1950204, 2019 · Zbl 1480.20062 · doi:10.1142/S0219498819502049 |
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