On Some Results in the Theory of Finite Partially Soluble Groups

Abstract

This article provides an overview of some recent results and ideas related to the study of finite groups depending on the restrictions on some systems of their sections. In particular, we discuss some properties of the lattice of all subgroups of a finite group related with conditions of permutability and generalized subnormality for subgroups. The paper contains more than 30 open problems which were posed, at different times, by some mathematicians working in the discussed direction.

Keywords

Finite group / Subgroup lattice / G-covering subgroup system / $\Pi $-subnormal subgroup')">$\Pi $-subnormal subgroup / $\sigma $-nilpotent group')">$\sigma $-nilpotent group

References

[1] Agrawal RK. Finite groups whose subnormal subgroups permute with all Sylow subgroups. Proc. Am. Math. Soc.. 1975, 47 77-83

[2] Asaad M, Heliel AA. On permutable subgroups of finite groups. Arch. Math.. 2003, 80 113-118

[3] Ballester-Bolinches, A., Esteban-Romero, R., Heliel, A.A., Almestadi, M.O.: ${\cal Z}$-permutable subgroups of finite groups, Preprint

[4] Ballester-Bolinches A, Esteban-Romero R. On finite soluble groups in which Sylow permutability is a transitive relation. Acta Math. Hung.. 2003, 101 193-202

[5] Ballester-Bolinches A, Ezquerro LM. Classes of Finite Groups. 2006 Dordrecht: Springer

[6] Ballester-Bolinches A, Guo X. Some results on $p$-nilpotency and solubility of finite groups. J. Algebra. 2000, 228 491-496

[7] Ballester-Bolinches A, Wang Y, Guo X. $c$-supplemented subgroups of finite groups. Glasgow Math J.. 2000, 42 383-389

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

[9] Ballester-Bolinches A, Ezquerro LM, Skiba AN. Local embeddings of some families of subgroups of finite groups. Acta Math. Sin.. 2009, 25 869-882

[10] Barnes DW, Kegel OH. Gaschütz functors on finite soluble groups. Math. Z.. 1966, 94 134-142

[11] Bianchi M, Mauri AG, Hauck P. On finite groups with nilpotent Sylow-normalizers. Arch. Math.. 1986, 47 193-197

[12] Curzio M. Sui sottogruppi di composizione dei gruppi finiti. Res. Mat.. 1958, 7 265-280

[13] Deskins WE. A condition for the solvability of a finite group. Illinois J. Math.. 1961, 2 306-313

[14] Deskins WE. On quasinormal subgroups of finite groups. Math. Z.. 1963, 82 125-132

[15] Doerk K, Hawkes T. Finite Soluble Groups. 1992 Berlin: Walter de Gruyter

[16] Ebert G, Bauman S. A note on subnormal and abnormal chains. J. Algebra. 1975, 36 2 287-293

[17] Fattahi A. Groups with only normal and abnormal subgroups. J. Algebra. 1974, 28 1 15-19

[18] Fedri V, Serens L. Finite soluble groups with supersoluble Sylow normalizers. Arch. Math.. 1988, 50 11-18

[19] Förster P. Finite groups all of whose subgroups are $\mathfrak{F}$-subnormal or $\mathfrak{F}$-subabnormal. J. Algebra. 1986, 103 1 285-293

[20] Guo W, Huang J, Skiba AN. On $G$-covering subgroup systems for some saturated formations of finite groups. Commun. Algebra. 2013, 41 2948-2956

[21] Guo W, Shum KP, Skiba AN. $G$-covering subgroup systems for the classes of supersoluble and nilpotent groups. Israel J. Math.. 2003, 138 125-138

[22] Guo W, Shum KP, Skiba AN. $X$-semipermutable subgroups of finite groups. J. Algebra. 2007, 315 31-41

[23] Guo, W., Skiba, A.N.: Finite groups whose $n$-maximal subgroups are $\sigma $-subnormal, Preprint, (2015)

[24] Guo, W., Skiba, A.N.: Finite groups with permutable Hall subgroups, submitted

[25] Guo, W., Skiba, A.N.: Gradewise properties of subgroups and its applications, submitted

[26] Guo W, Skiba AN. On $G$-covering subgroup systems of finite groups. Acta Math. Hung.. 2011, 133 4 376-386

[27] Guo, W., Skiba, A.N.: On $\Pi $-permutable subgroups of finite groups, submited

[28] Guo, W., Skiba, A.N.: On $\sigma $-supersoluble groups and one generalization of $CLT$-groups, submitted

[29] Guo W. Finite groups with given indices of normalizers of Sylow subgroups. Sib. Math. J.. 1996, 37 2 253-258

[30] Guo W. On $\mathfrak{F}$-supplemented subgroups of finite groups. Manuscripta Math.. 2008, 127 139-150

[31] Guo W. Structure Theory for Canonical Classes of Finite Groups. 2015 Heidelberg: Springer

[32] Guo X, Wang J, Shum KP. On semi-cover-avoiding maximal subgroups and solvability of finite groups. Commun. Algebra. 2006, 34 3235-3244

[33] Guo XY, Shum KP. Cover-avoidance properties and the structure of finite groups. J. Pure Appl. Algebra. 2003, 181 297-308

[34] Guo W, Shum KP, Skiba AN. $G$-covering subgroup systems for the classes of $p$-supersoluble and $p$-nilpotent finite groups. Sib. Math. J.. 2004, 45 3 442-453

[35] Guo W, Skiba AN. Gradewise properties of subgroups of finite groups. Sib. Math. J.. 2015, 56 3 384-392

[36] Guo W, Skiba AN. Finite groups with generalized Ore supplement conditions for primary subgroups. J. Algebra. 2015, 432 205-227

[37] Hall P. A characteristic property of soluble groups. J. Lond. Math. Soc.. 1937, 12 205-221

[38] Huppert B. Endliche Gruppen I. 1967 Berlin: Springer

[39] Huppert B. Normalteiler und maximale Untergruppen endlicher Gruppen. Math. Z.. 1954, 60 409-434

[40] Huppert B. Zur Sylowstruktur Auflosbarer Gruppen. II. Arch. Math.. 1964, 15 251-257

[41] Isaacs IM. Semipermutable $\pi $-subgroups. Arch. Math.. 2014, 102 1-6

[42] Iwasawa K. Über die endlichen Gruppen und die Verbände ihrer Untergruppen. J. Fac. Sci. Imp. Univ. Tokyo. Sect.. 1941, 1 4 171-199

[43] Johnson DL. A note on supersoluble groups. Can. J. Math.. 1971, 23 562-564

[44] Kegel OH. Sylow-Gruppen and Subnormalteilerendlicher Gruppen. Math. Z.. 1962, 78 205-221

[45] Kegel OH. Untergruppenverbande endlicher Gruppen, die den subnormalteilerverband each enthalten. Arch. Math.. 1978, 30 3 225-228

[46] Kimber T. Modularity in the lattice of $\Sigma $-permutable subgroups. Arch. Math.. 2004, 83 193-203

[47] Kleidman PB. A proof of the Kegel-Wielandt conjecture on subnormal subgroups. Ann. Math.. 1977, 133 369-428

[48] Lennox JC, Stonehewer SE. Subnormal Subgroups of Groups. 1987 Oxford: Clarendon Press

[49] Li B. On $\Pi $-property and $\Pi $-normality of subgroups of finite groups. J. Algebra. 2011, 334 321-337

[50] Lu, J., Meng, W.: Finite groups with non-subnormal subgroups, Commun. Algebra (submitted)

[51] Lu J, Meng W. On solvability of finite groups with few non-normal subgroups. Commun. Algebra. 2015, 43 5 1752-1756

[52] Mal’cev AI. Algebraic Systems. 1970 Main Editorial Board for Physical and Mathematical Literature, Moscow: Nauka

[53] Mann A. Fnite groups whose $n$-maximal subgroups are subnormal. Trans Am. Math. Soc.. 1968, 132 395-409

[54] Maslova NV, Revin DO. Finite groups whose maximal subgroups have the hall property. Sib. Adv. Math.. 2012, 23 3 196-209

[55] Maslova NV. Non-abelian composition factors of a finite group whose all maximal subgroups are Hall. Sib. Math. J.. 2012, 53 5 853-861

[56] Monakhov VS, Knyagina VN. On finite groups with some subnormal Schmidt subgroups. Sib. Math. J.. 2004, 45 6 1316-1322

[57] Monakhov, V.S., Sokhor, I.L.: Finite soluble groups with nilpotent wide subgroups, ArXiv. org e-Print archive, arXiv:1603.06551v1 [math. GR], 21 (Mar 2016)

[58] Monakhov VS. Finite $\pi $-solvable groups whose maximal subgroups have the Hall property. Math. Notes. 2008, 84 3 363-366

[59] Ore O. Contributions in the theory of groups of finite order. Duke Math. J.. 1939, 5 431-460

[60] Ore O. Structures and group theory. Duke Math. J.. 1938, 4 247-269

[61] Robinson DJS. The structure of finite groups in which permutability is a transitive relation. J. Austral. Math. Soc.. 2001, 70 143-159

[62] Schmid P. Subgroups permutable with all Sylow subgroups. J. Algebra. 1998, 207 285-293

[63] Schmidt R. Subgroup Lattices of Groups. 1994 Berlin: Walter de Gruyter

[64] Semenchuk VN, Skiba AN. On one generalization of finite $\mathfrak{U}$-critical groups. J. Algebra Appl.. 1916, 15 4 21-36

[65] Semenchuk, V.N.: Finite groups with a system of minimal non-$\mathfrak{F}$-groups, In: Subgroup structure of finite groups. Nauka i Tehnika, Minsk, pp. 138–139 (1981)

[66] Semenchuk VN. Finite groups with $f$-abnormal and $f$-subnormal subgroups. Mat. Zametki. 1994, 55 6 111-115

[67] Semenchuk, V.N.: The structure of finite groups with the $\mathfrak{F}$-abnormal or $\mathfrak{F}$-subnormal subgroups. “Questions of Algebra”, Publishing House “University” Minsk 2, 50–55 (1986)

[68] Shemetkov LA. Formations of finite groups. 1978 Nauka, Main Editorial Board for Physical and Mathematical Literature: Moscow

[69] Skiba AN. A generalization of a Hall theorem. J. Algebra Appl.. 1915, 15 4 21-36

[70] Skiba AN. Finite groups with given systems of generalized permutable subgroups. Proc. Francisk Skorina Gomel State Univ.. 2006, 36 3 12-31

[71] Skiba, A.N.: On the lattice of all $\Pi $-subnormal subgroups of finite groups, Preprint, (2016)

[72] Skiba AN. On weakly $s$-permutable subgroups of finite groups. J. Alegebra. 2007, 315 192-209

[73] Skiba AN. On $\sigma $-properties of finite groups I. Probl. Phys. Math. Tech.. 2014, 4 21 89-96

[74] Skiba AN. On $\sigma $-subnormal and $\sigma $-permutable subgroups of finite groups. J. Algebra. 2015, 436 1-16

[75] Spencer AE. Maximal nonnormal chains in finite groups. Pacific J. Math.. 1968, 27 167-173

[76] Tihonenko TV, Tutyanov VN. Finite groups with maximal Hall subgroups. Izv. F. Skorina Gomel State Univ.. 2008, 50 5 198-206

[77] Vedernikov VA. Finite groups in which every nonsolvable maximal subgroup is a Hall subgroupl. Proc. Steklov Inst. Math.. 2014, 285 1 191-202

[78] Vedernikov VA. Finite groups with subnormal Schmidt subgroups. Algebra Logica. 2007, 46 6 669-687

[79] Vorob’ev NT, Zagurski VN. Fitting classes with the given properties of Hall subgroups. Math. Zametki. 2005, 78 2 234-240

[80] Wang Y. Finite groups with some subgroups of Sylow subgroups c-supplemented. J. Algebra. 2000, 224 467-478

[81] Weinstein M. Between Nilpotent and Solvable. 1982 Passaic: Polygonal Publishing House

[82] Wielandt H. Eine Verallgemenerung der invarianten Untergruppen. Math. Z.. 1939, 45 200-244

[83] Wielandt H. Kriterion für Subnormalität in endlichen Gruppen. Math. Z.. 1974, 138 199-203

[84] Zacher G. Sui gruppi finiti per cui il reticolo dei sottogruppi di composizione e destributivo. Rend. Sem. Mat. Univ. Padova. 1957, 27 75-79

[85] Zappa G. Sui gruppi finiti per cui il reticolo dei sottogruppi di composizione e modulare. Boll. Un. Mat. Ital.. 1956, 11 3 315-318

[86] Zhang J. Sylow numbers of finite groups. J. Algebra. 1995, 176 1 111-123