On $\sigma$-properties of finite groups~II

A. N. Skiba

Problemy fiziki, matematiki i tehniki, no. 3 (2015), pp. 70-83.

Résumé

Let $G$ be a finite group, $\sigma=\{\sigma_i \mid i\in I\}$ some partition of the set $\mathbb{P}$ of all primes and $\Pi$ a subset of the set $\sigma$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\Pi$-set of $G$ if $\mathcal{H}$ contains exact one Hall $\sigma_i$-subgroup of $G$ for every $\sigma_i\in\Pi$ such that $\sigma_i\cap\pi(G)\ne\varnothing$. We say also that $G$ is: $\Pi$-full if $G$ possess a complete Hall $\Pi$-set; a $\Pi$-full group of Sylow type if for each $\sigma_i\in\Pi$, every subgroup $E$ of $G$ is a $D_{\sigma_i}$-group, that is, $E$ has a Hall $\sigma_i$-subgroup $H$ and every $\sigma_i$-subgroup of $E$ is contained in some conjugate of $H^x$ ($x\in E$). In this paper we study properties of finite $\Pi$-full groups. The work continues the research of the paper [1].

Détail

Keywords: finite group, $\Pi$-full group, $\sigma$-nilpotent group
Mots-clés : $\sigma$-soluble group, $\sigma$-quasinilpotent group.

Bibliographie

[1] A. N. Skiba, “On $\sigma$-properties of finite groups I”, Problems of Physics, Mathematics and Technics, 2014, no. 4(21), 89–96 | Zbl

[2] A. N. Skiba, “On $\sigma$-subnormal and $\sigma$-permutable subgroups of finite groups”, J. Algebra, 436 (2015), 1–16 | DOI | MR | Zbl

[3] A. N. Skiba, “A generalization of a Hall theorem”, J. Algebra and its Application, 2015 (to appear)

[4] B. Huppert, N. Blackburn, Finite groups, v. III, Springer-Verlag, Berlin–Heidelberg–New York, 1982

[5] K. Doerk, T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin–New York, 1992 | MR

[6] W. Guo, K. P. Shum, A. N. Skiba, “$X$-semipermutable subgroups of finite groups”, J. Algebra, 315 (2007), 31–41 | DOI | MR | Zbl

[7] A. N. Skiba, On some properties of finite $\sigma$-soluble and $\sigma$-nilpotent groups, Preprint, 2014

[8] W. Guo, A. N. Skiba, “Finite groups with permutable complete Wielandt sets of subgroups”, J. Group Theory, 18 (2015), 191–200 | DOI | MR | Zbl

[9] J. Zhang, “Sylow numbers of finite groups”, J. Algebra, 176 (1995), 111–123 | DOI | MR | Zbl

[10] W. Guo, “Finite groups with given indices of normalizers of Sylow subgroups”, Siberian Math. J., 37 (1996), 207–214 | DOI | MR

[11] B. Huppert, Endliche Gruppen, v. I, Springer-Verlag, Berlin–Heidelberg–New York, 1967 | MR | Zbl

[12] T. M. Gagen, Topics in finite groups, London Math. Soc. Lectures Note Series, 16, Cambridge Univ. Press, London, 1976 | MR | Zbl

[13] A. N. Skiba, L. A. Shemetkov, “Multiply $\mathcal{L}$-Composition Formations of Finite Groups”, Ukrainsk. Math. Z., 52:6 (2000), 783–797 | MR | Zbl

[14] L. A. Shemetkov, Formations of finite groups, Nauka. Main Editorial Board for Physical and Mathematical Literature, M., 1978 | MR | Zbl

[15] W. Guo, A. N. Skiba, “On $\mathfrak{F}\phi^*$-hypercentral subgroups of finite groups”, J. Algebra, 372 (2012), 275–292 | DOI | MR | Zbl

[16] H. Bender, “On groups with abelian Sylow 2-subgroups”, Math. Z., 117 (1970), 164–176 | DOI | MR | Zbl

[17] D. Gorenstein, Finite Groups, Harper Row Publishers, New York–Evanston–London, 1968 | MR | Zbl

[18] V. S. Monakhov, O. A. Shpyrko, “The nilpotent $\pi$-length of maximal Subgroups in finite $\pi$-soluble groups”, Moscow University Mathematics Bulletin, 64:6 (2009), 229–234 | DOI | MR | Zbl

[19] O. H. Kegel, “Sylow-Gruppen und Subnormalteiler endlicher Gruppen”, Math. Z., 78 (1962), 205–221 | DOI | MR | Zbl

[20] A. N. Skiba, On $\Pi$-permutable subgroups of finite groups, Preprint, 2015 | MR

[21] I. M. Isaacs, “Semipermutable $\pi$-subgroups”, Arch. Math., 102 (2014), 1–6 | DOI | MR | Zbl

[22] W. E. Deskins, “On quasinormal subgroups of finite groups”, Math. Z., 82 (1963), 125–132 | DOI | MR | Zbl

[23] P. Schmid, “Subgroups permutable with all Sylow subgroups”, J. Algebra, 207 (1998), 285–293 | DOI | MR | Zbl

[24] R. K. Agrawal, “Finite groups whose subnormal subgroups permute with all Sylow subgroups”, Proc. Amer. Math. Soc., 47 (1975), 77–83 | DOI | MR | Zbl

[25] D. J. S. Robinson, “The structure of finite groups in which permutability is a transitive relation”, J. Austral. Math. Soc., 70 (2001), 143–159 | DOI | MR | Zbl

[26] R. A. Brice, J. Cossey, “The Wielandt subgroup of a finite soluble groups”, J. London Math. Soc., 40 (1989), 244–256 | DOI | MR

[27] J. C. Beidleman, B. Brewster, D. J. S. Robinson, “Criteria for permutability to be transitive in finite groups”, J. Algebra, 222 (1999), 400–412 | DOI | MR | Zbl

[28] A. Ballester-Bolinches, R. Esteban-Romero, “Sylow permutable subnormal subgroups”, J. Algebra, 251 (2002), 727–738 | DOI | MR | Zbl

[29] A. Ballester-Bolinches, J. C. Beidleman, H. Heineken, “Groups in which Sylow subgroups and subnormal subgroups permute”, Illinois J. Math., 47 (2003), 63–69 | MR | Zbl

[30] A. Ballester-Bolinches, J. C. Beidleman, H. Heineken, “A local approach to certain classes of finite groups”, Comm. Algebra, 31 (2003), 5931–5942 | DOI | MR | Zbl

[31] M. Asaad, “Finite groups in which normality or quasinormality is transitive”, Arch. Math., 83:4 (2004), 289–296 | DOI | MR | Zbl

[32] A. Ballester-Bolinches, J. Cossey, “Totally permutable products of finite groups satisfying SC or PST”, Monatsh. Math., 145 (2005), 89–93 | DOI | MR

[33] K. Al-Sharo et al., “Some characterizations of finite groups in which semipermutability is a transitive relation”, Forum Math., 22 (2010), 855–862 | DOI | MR | Zbl

[34] J. C. Beidleman, M. F. Ragland, “Subnormal, permutable, and embedded subgroups in finite groups”, Central Eur. J. Math., 9:4 (2011), 915–921 | DOI | MR | Zbl

[35] X. Yi, A. N. Skiba, “Some new characterizations of PST-groups”, J. Algebra, 399 (2014), 39–54 | DOI | MR | Zbl

[36] R. Schmidt, Subgroup lattices of groups, Walter de Gruyter, Berlin–New York, 1994 | MR | Zbl

[37] A. Ballester-Bolinches, R. Esteban-Romero, “On finite soluble groups in which Sylow permutability is a transitive relation”, Acta Math. Hungar., 101 (2003), 193–202 | DOI | MR | Zbl

[38] A. N. Skiba, “On weakly $s$-permutable subgroups of finite groups”, J. Algebra, 315 (2007), 192–209 | DOI | MR | Zbl

[39] L. A. Shemetkov, A. N. Skiba, “On the $\mathcal{X}\Phi$-hypercentre of finite groups”, J. Algebra, 322 (2009), 2106–2117 | DOI | MR | Zbl

[40] W. Guo, A. N. Skiba, “Finite groups with given $s$-embedded and $n$-embedded subgroups”, J. Algebra, 321 (2009), 2843–2860 | DOI | MR | Zbl

[41] X. Yi, “Finite groups with cyclic $S$-cofactors of subgroups”, J. Algebra Appl., 14 (2015), 1–9 | MR

[42] S. Li, J. Liu, “On Hall subnormally embedded and generalized nilpotent groups”, J. Algebra, 388 (2013), 1–9 | DOI | MR | Zbl

[43] J. Liu, S. Li, “$CLT$-groups with Hall $S$-quasinormally embedded subgroups”, Ukrainian Math. J., 66:8 (2015), 1281–1288 | DOI | MR

[44] A. N. Skiba, The $\sigma$-permutable closure and the $\sigma$-core of subgroups, Preprint, 2015 | MR

[45] A. Ballester-Bolinches, R. Esteban-Romero, M. Asaad, Products of Finite Groups, Walter de Gruyter, Berlin–New York, 2010 | MR | Zbl

[46] Ya. G. Berkovich, “Finite groups with big kernels of maximal subgroups”, Siberian Math. J., 9 (1968), 606–610

[47] G. Cutolo, E. I. Khukhro, J. C. Lennoks, J. Wiegold, S. Rinauro, H. Smith, “Finite core-$p$-groups”, J. Algebra, 188:2 (1997), 701–719 | DOI | MR | Zbl

[48] J. Poland, “On finite groups whose subgroups have simple core factors”, Proc. Japan Acad., 47 (1971), 606–610 | DOI | MR | Zbl

[49] L. P. Avdashkova, S. F. Kamornikov, “On a class of groups with given cofactors of maximal subgroups”, Mathematical Notes, 87:5/6 (2010), 643–649 | MR | Zbl

[50] I. V. Lemeshev, V. S. Monakhov, “The solvability criteria for finite groups with restrictions on cofactors of maximal sugroups”, Problems of Physics, Mathematics and Technics, 2:11 (2012), 88–94 | Zbl

[51] Y. Liu, X. Yi, “Finite groups in which primary subgroups have cyclic cofactors”, Bull. Malaysian Math. Sciences Soc., 34:2 (2011), 337–344 | MR | Zbl

[52] A. Ballester-Bolinches, M. C. Pedraza-Aguilera, “Sufficient conditions for supersolvebility of finite groups”, J. Pure Appl. Algebra, 127 (1998), 113–118 | DOI | MR | Zbl

[53] S. Srinivasan, “Two sufficient conditions for supersolvability of finite groups”, Israel J. Math., 35:3 (1980), 210–214 | DOI | MR | Zbl

[54] G. L. Walls, “Groups with maximal subgroups of Sylow subgroups normal”, Israel J. Math., 43:2 (1982), 166–168 | DOI | MR | Zbl

[55] P. B. Kleidman, “A proof of the Kegel–Wielandt conjecture on subnormal subgroups”, Ann. Math., 133 (1977), 369–428 | DOI | MR

[56] J. C. Lennox, S. E. Stonehewer, Subnormal Subgroups of Groups, Clarendon Press, Oxford, 1987 | MR | Zbl

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

[58] V. A. Vedernikov, “On $\pi$-properties of finite groups”, Arithmetic and Subgroup Structure of Finite Groups, Nauka i Tehnika, Mn., 1986, 13–19 | MR