Semipermutable subgroups and s-semipermutable subgroups in finite groups

Abstract

Suppose that H is a subgroup of a finite group G. We call H is semipermutable in G if HK = KH for any subgroup K of G such that (|H|, |K|) = 1; H is s-semipermutable in G if HG_p = G_pH, for any Sylow p-subgroup G_p of G such that (|H|, p) = 1. These two concepts have been received the attention of many scholars in group theory since they were introduced by Professor Zhongmu Chen in 1987. In recent decades, there are a lot of papers published via the application of these concepts. Here we summarize the results in this area and gives some thoughts in the research process.

Key words： Semipermutable subgroup; s-semipermutable subgroup; maximal subgroup; minimal subgroup; the generalized Fitting-subgroup; formation

References

1 Al-sharo D M , Sulaiman H . On some relations of subnormal subgroups and semipermutability of a finite group. In: Proceedings of the 21st National Symposium on Mathematical Sciences (SKSM21), AIP Conf. Proc., Vol. 1605, Melville, NY: AIP Publishing, 2014, 628- 632

2 Al-sharo K A , Beidleman J C , Heineken H , Ragland M F . Some characterizations of finite groups in which semipermutability is a transitive relation. Forum Math., 2010, 22 (5): 855- 862

3 Asaad M . On maximal subgroups of Sylow subgroups of finite groups. Comm. Algebra, 1998, 26 (11): 3647- 3652

4 Asaad M . Finite groups with given nearly s-embedded subgroups. Acta Math. Hungar., 2014, 144 (2): 499- 514

5 Asaad M , Csörgö P . The influence of minimal subgroups on the structure of finite groups. Arch. Math., 1999, 72: 401- 404

6 Asaad M , Ramadan M , Shaalan A . The influence of π-quasinormality of maximal subgroups of Sylow subgroups of Fitting subgroups of a finite group. Arch. Math., 1991, 56: 521- 527

7 Ballester-Bolinches A , Beidleman J C , Esteban-Romero R , Ragland M F . On a class of supersoluble groups. Bull. Aust. Math. Soc., 2014, 90 (2): 220- 226

8 Ballester-Bolinches A , Esteban-Romero R , Asaad M . Products of Finite Groups. New York: Walter de Gruyter, 2010

9 Ballester-Bolinches A , Esteban-Romero R , Qiao S . A note on a result of Guo and Isaacs about p-supersolubility of finite group. Arch. Math., 2016, 106: 501- 506

10 Ballester-Bolinches A , Ezquerro L M , Skiba A N . Local embeddings of some families of subgroups of finite groups. Acta Math. Sin. (Engl. Ser.), 2009, 25: 869- 882

11 Ballester-Bolinches A , Li Y M , Su N , Xie Z . On π-S-permutable subgroups of finite groups. Mediterr. J. Math., 2016, 13 (1): 93- 99

12 Ballester-Bolinches A , Wang Y , Guo X Y . c-supplemented subgroups of finite groups. Glasgow Math. J., 2000, 42: 383- 389

13 Beidleman J C , Ragland M F . Subnormal, permutable and embedded subgroups in finite groups. Cent. Eur. J. Math., 2011, 9 (4): 915- 921

14 Berkovich Y , Isaacs I M . p-supersolvability and actions on p-groups stabilizing certain subgroups. J. Algebra, 2014, 414: 82- 94

15 Buckley J . Finite groups whose minimal subgroups are normal. Math. Z., 1970, 116: 15- 17

16 Chen X , Guo W B . On weakly S-embedded and weakly τ-embedded subgroups. Sib. Math. J., 2013, 54 (5): 931- 945

17 Chen Z M . Generalization of the Schur-Zassenhaus theorem. J. Math., 1998, 18 (3): 290- 294

18 Chen Z M . Inner and Outer Σ-groups and Minimal Non Σ-groups, Chongqing: Southwest Normal University Press, 1988 (in Chinese)

19 Chen Z M . On a theorem of Srinivasan. Southwest Normal Univ. Nat. Sci., 1987, 12 (1): 1- 4 (in Chinese)

20 Dedekind R . Über Gruppen, deren sámtliche Teiler Normalteiler sind, Math. Ann., 1897, 48: 548- 561

21 Deskin W E . On quasinormal subgroups of finite groups. Math. Z., 1963, 82: 125- 132

22 Doerk K , Hawkes T . Finite Soluble Groups, De Gruyter Expositions in Mathematics, Vol. 4. Berlin: Walter de Gruyter, 1992

23 Gorenstein D . Finite Groups. New York: Chelsea, 1968

24 Guo W B . On F-supplemented subgroups of finite groups. Manuscripta Math., 2008, 127: 139- 150

25 Guo X Y , Zhao X H . π-quasinormality of the maximal subgroups of a Sylow subgroup in a local subgroup. Acta Math. Sci. (Chin. Ser.), 2008, 28 (6): 1222- 1226 (in Chinese)

26 Guo Y H , Isaacs I M . Conditions on p-subgroups implying p-nilpotence or p-supersolvability. Arch. Math., 2015, 105: 215- 222

27 Hall P . On a theorem of Frobenius. Proc. London Math. Soc., 1936, 40: 468- 501

28 Heliel A A , Alharbia S M . The infuence of certain permutable subgroups on the structure of finite groups. Internat. J. Algebra, 2010, 4: 1209- 1218

29 Huang Y J , Li Y M . A local version of a result of Chen. J. Math., 2013, 33 (4): 584- 590

30 Huppert B . Endliche Gruppen I. Grund. Math Wiss Vol 134, Berlin-Heidelberg-New York: Springer-Verlag, 1967 (in German)

31 Huppert B , Blackburn N . Finite Groups III. Berlin: Springer-Verlag, 1982

32 Isaacs I M . Semipermutable π-subgroups. Arch. Math., 2014, 102 (1): 1- 6

33 Kegel O H . Sylow-Gruppen und Subnormalteiler endlicher Gruppen. Math Z, 1962, 78: 205- 221 (in German)

34 Kurzwell H , Stellmacher B . The Theory of Finite Groups: An Introduction. New York: Springer, 2003

35 Li B J . On Π-property and Π-normality of subgroups of finite groups. J. Algebra, 2011, 334 (1): 321- 337

36 Li S , He X . On normally embedded subgroups of prime power order in finite groups. Comm. Algebra, 2008, 36: 2333- 2340

37 Li Y M , He X L , Wang Y M . On s-semipermutable subgroups of finite groups. Acta Math. Sin. (Engl. Ser.), 2010, 26 (11): 2215- 2222

38 Li Y M , Li B J . On minimal weakly s-supplemented subgroups of finite groups. J. Algebra Appl., 2011, 10: 811- 820

39 Li Y M , Miao L Y . p-hypercyclically embedding and Π-property of subgroups of finite groups. Comm. Algebra, 2017, 45 (8): 3468- 3474

40 Li Y M , Qiao S H . On weakly s-normal subgroups of finite groups. Ukrainian Math. J., 2012, 63 (11): 1770- 1780

41 Li Y M , Qiao S H , Su N , Wang Y M . On weakly s-semipermutable subgroups of finite groups. J. Algebra, 2012, 371: 250- 261

42 Li Y M , Qiao S H , Wang Y M . A note on a result of Skiba. Sib. Math. J., 2009, 50 (3): 467- 473

43 Li Y M , Wang Y M . The influence of minimal subgroups on the structure of finite group. Proc. Amer. Math. Soc., 2003, 131 (2): 337- 349

44 Li Y M , Wang Y M , Wei H Q . The influence of π-quasinormality of maximal subgroups of Sylow subgroups of a finite group. Arch. Math. (Basel), 2003, 81 (3): 245- 252

45 Lu J K , Li S R . On s-semipermutable subgroups of finite groups. J. Math. Res. Exposition, 2009, 29 (6): 985- 991

46 Lukyanenko V O , Skiba A N . On weakly τ-quasinormal subgroups of finite groups. Acta Math. Hungar., 2009, 125 (3): 237- 248

47 Lukyanenko V O , Skiba A N . Finite groups in which τ-quasinormality is a transitive relation. Rend. Semin. Mat. Univ. Padova, 2010, 124: 231- 246

48 Maier R , Schmid P . The embedding of quasinormal subgroups in finite groups. Math. Z., 1973, 131 (3): 269- 272

49 Mao Y M , Mahboob A , Guo W B . S-semiembedded subgroups of finite groups. Front. Math. China, 2015, 10 (6): 1401- 1413

50 Mazurov V D , Khukhro E I (eds.), Unsolved Problems in Group Theory, 15th Ed.. The Kourovka Notebook, Novosibirsk: Inst. Math. of Russian Acad. Sci. Sib. Div., 2002

51 Miao L Y , Ballester-Bolinches A , Esteban-Romero R , Li Y M . On the supersoluble hypercentre of a finite group. Monatsh. Math., 2017, 184: 641- 648

52 Miao L Y , Li Y M . Some criteria for p-supersolvability of a finite group. Comm. Math. Stat., 2017, 5 (3): 339- 348

53 Obaid M . Finite groups whose certain subgroups of prime power order are S-semipermutable. ISRN Algebra, 2011: 339- 348

54 Ore O . Structures of group theory I. Duke Math. J., 1937, 3: 149- 174

55 Qiu Z , Qiao S . s-semipermutability of subgroups of p-nilpotent residual and p-supersolubility of a finite group. J. Algebra Appl., 2021, 20: 2150117

56 Ramadan M . Influence of normality on maximal subgroups of Sylow subgroups of a finite group. Acta Math. Hungar., 1992, 59 (1/2): 107- 110

57 Ren Y C . Notes on π-quasi-normal subgroups in finite groups. Proc. Amer. Math. Soc., 1993, 117: 631- 636

58 Schmid P . Subgroups permutable with all Sylow subgroups. J. Algebra, 1998, 207: 285- 293

59 Sergienko V I . A criterion for the p-solubility of finite groups. Mat. Zametki, 1971, 9: 375- 383

60 Shemetkov L , Skiba A . On the χφ-hypercenter of finite groups. J. Algebra, 2009, 322: 2106- 2117

61 Shen Z C , Zhang J S , Wu S L . Finite groups with weakly s-semipermutably embedded subgroups. Intern. Elect. J. Algebra, 2012, 11: 111- 124

62 Skiba A N . On weakly s-permutable subgroups of finite groups. J. Algebra, 2007, 315: 192- 209

63 Srinivasan S . Two sufficient conditions for supersolvability of finite groups. Israel J. Math., 1980, 35 (3): 210- 214

64 Stonehewer S E . Permutable subgroups of infinite groups. Math. Z., 1972, 125: 1- 16

65 Su N , Li Y M , Wang Y M . A criterion of p-hypercyclically embedded subgroups of finite groups. J. Algebra, 2014, 400: 82- 93

66 Thompson J G . An example of core-free quasinormal subgroups of p-groups. Math. Z., 1967, 96: 226- 226

67 Wang L F . The influence of s-semipermutable subgroups on the p-supersolvability of finite groups. J. Math. Stud., 2009, 42 (4): 434- 440 (in Chinese)

68 Wang L F , Li Y M , Wang Y M . Finite groups in which (S-)semipermutability is a transitive relation. Intern. J. Algebra, 2008, 2 (3): 143- 152

69 Wang L F , Wang Y M . On s-semipermutable maximal and minimal subgroups of Sylow p-subgroups of finite groups. Comm. Algebra, 2006, 34 (1): 143- 149

70 Wang L F , Zhang Q H . Influence of s-semipermutability of some subgroups of prime power order on structure of finite groups. J. Math. Res. Exposition., 2005, 25 (3): 423- 428

71 Wang Y M . C-normality of groups and its properties. J. Algebra, 1996, 180: 954- 965

72 Wei H Q , Wang Y M , Li Y M . On c-supplemented maximal and minimal subgroups of Sylow subgroups of finite groups. Proc. Amer. Math. Soc., 2004, 132 (8): 2197- 2204

73 Wei H Q , Dai Q , Zhang H , Lv Y , Yang L . On c^#-normal subgroups in finite groups, Front. Math. China, 2018, 13 (5): 1169- 1178

74 Wei H Q , Yang L , Dong S . Local c*-supplementation of some subgroups in finite groups, Comm. Algebra, 2016, 44 (11): 4986- 4994

75 Wu X , Li X . Weakly s-semipermutable subgroups and structure of finite groups. Comm. Algebra, 2020, 48 (6): 2307- 2314

76 Xu X Y , Li Y M . A criterion on the finite p-nilpotent groups. J. Math. Res. Appl., 2019, 39 (3): 254- 258

77 Xu Y , Li X H . Weakly s-semipermutable subgroups of finite groups. Front. Math. China, 2011, 6 (1): 161- 175

78 Yu H R . A note on S-semipermutable subgroups of finite groups. Rend. Semin. Mat. Univ. Padova, 2017, 138: 257- 263

79 Zhang Q H . s-semipermutability and abnormality in finite groups. Comm. Algebra, 1999, 27 (9): 4515- 4524

80 Zhang Q H , Wang L F . Finite non-abelian simple groups which contain a non-trivial semipermutable subgroup. Algebra Colloq., 2005, 12 (2): 301- 307

81 Zhang Q H , Wang L F . The influence of s-semipermutable subgroups on the structure of finite groups. Acta Math. Sinica (Chin. Ser.), 2005, 48 (1): 81- 88 (in Chinese)

82 Zhang Q H , Wang L F , Guo P F . The structure of some finite groups. Southeast Asian Bull. Math., 2006, 30: 995- 1002

83 Zhao T , Lu G F . The influence of partially s-embedded subgroups on the structure of a finite group. Acta Univ. Apulensis, 2014, 38: 197- 209