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

A. N. Skiba

Problemy fiziki, matematiki i tehniki, no. 1 (2016), pp. 52-62.

Résumé

Let $G$ be a finite group and $\sigma=\{\sigma_i\mid i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, that is, $\mathbb{P}=\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$. Let $\Pi\subseteq\sigma$. We say that a subgroup $A$ of $G$ is $\Pi$-subnormal in $G$ if there is a subgroup chain $A=A_0\leqslant A_1\leqslant\dots\leqslant A_t=G$ such that either $A_{i-1}$ is normal in $A_i$ or $A_i/(A_{i-1})_{A_i}$ is a $\sigma_j$-group for some $\sigma_j\in\Pi$ for all $i=1,\dots,t$. In this paper, we discuss properties of $\Pi$-subnormal subgroups and some other $\sigma$-properties of finite groups. The work continues the research in the papers [1]–[5].

Détail

Keywords: finite group, $\Pi$-subnormal subgroup, the lattice of the $\Pi$-subnormal subgroups, $\sigma$-supersoluble group
Mots-clés : $CLT_\sigma$-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] W. Guo, A. N. Skiba, “Finite groups with permutable complete Wielandt sets of subgroups”, J. Group Theory, 18 (2014), 191–200 | MR

[3] A. N. Skiba, “On $\sigma$-properties of finite groups II”, Problems of Physics, Mathematics and Technics, 2015, no. 3(24), 67–81

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

[5] A. N. Skiba, “A generalization of a Hall theorem”, J. Algebra and its Application, 15:4 (2015), 21–36 | MR

[6] O. H. Kegel, “Untergruppenverbande endlicher Gruppen, die den subnormalteilerverband each enthalten”, Arch. Math., 30:3 (1978), 225–228 | DOI | MR | Zbl

[7] A. Ballester-Bolinches, L. M. Ezquerro, Classes of Finite Groups, Springer-Verlag, Dordrecht, 2006 | MR | Zbl

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

[9] G. Zappa, “Sui gruppi finiti per cui il reticolo dei sottogruppi di composizione e modulare”, Bull. Un. Mat. Ital., 11:3 (1956), 315–318 | MR | Zbl

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

[11] G. Zacher, “Sui gruppi finiti per cui il reticolo dei sottogruppi di composizione e destributivo”, Rend. Sem. Mat. Univ. Padova, 27 (1957), 75–79 | MR | Zbl

[12] M. Curzio, “Sui sottogruppi di composizione dei gruppi finiti”, Reserche Mat., 7 (1958), 265–280 | MR | Zbl

[13] M. Weinstein (ed.), Between Nilpotent and Solvable, Polygonal Publishing House, Passaic N.J., 1982 | MR | Zbl

[14] A. Ballester-Bolinches, J. C. Beidlemann, R. Esteban-Romero, “On some classes of supersoluble groups”, J. Algebra, 312 (2007), 445–454 | DOI | MR | Zbl

[15] S. Li, J. He, G. Nong, L. Zhou, “On Hall normally embedded subgroups of finite groups”, Comm. Algebra, 37 (2009), 3360–3367 | DOI | MR | Zbl

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

[17] J. Liu, S. Li, J. He, “$CLT$-groups with normal or subnormal subgroups”, J. Algebra, 382 (2012), 99–106 | DOI

[18] J. Lio, S. Li, “$CLT$-groups with Hall $S$-quasinormally embedded subgroups”, Ukrain. Math. Journal, 66 (2014), 1281–1287 | DOI | MR

[19] J. F. Humphreys, “On goups satifying the converse Lagrang's theorem”, Proc. Camb. Phil. Soc., 75 (1974), 25–32 | DOI | MR | Zbl

[20] O. Ore, “Contributions to the theory of finite groups”, Duke. Math. J., 5 (1940), 431–460 | DOI | MR

[21] G. Zappa, “Remark on a recent paper of O. Ore”, Duke. Math. J., 6 (1940), 511–512 | DOI | MR

[22] W. Guo, A. N. Skiba, On $\sigma$-supersoluble groups and one generalization of $CLT$-groups, Preprint, 2015

[23] W. Guo, A. N. Skiba, On $H_\sigma$-subnormally embedded and $H_\sigma$-permutably embedded subgroups of finite groups, Preprint, 2015

[24] A. Ballester-Bolinches, Shou Homg Qiao, “On a problem proposed by S. Li and J. Liu”, Arch. Math., 102 (2014), 109–111 | DOI | MR | Zbl

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

[26] B. Huppert, “Normalteiler und maximale Untergruppen endlicher Gruppen”, Math. Z., 60 (1954), 409–434 | DOI | MR | Zbl

[27] A. Mann, “Fnite groups whose $n$-maximal subgroups are subnormal”, Trans. Amer. Math. Soc., 132 (1968), 395–409 | MR | Zbl

[28] W. E. Deskins, “A condition for the solvability of a finite group”, Illinois J. Math., 2 (1961), 306–313 | MR

[29] A. E. Spencer, “Maximal nonnormal chains in finite groups”, Pacific J. Math., 27 (1968), 167–173 | DOI | MR | Zbl

[30] W. Guo, H. V. Legchekova, A. N. Skiba, “The structure of finite non-nilpotent groups in which every 2-maximal subgroup permutes with all 3-maximal subgroups”, Comm. Algebra, 37 (2009), 2446–2456 | DOI | MR | Zbl

[31] W. Guo, H. V. Legchekova, A. N. Skiba, “Finite groups in which every 3-maximal subgroup permutes with all maximal subgroups”, Mathematical Notes, 86 (2009), 325–332 | DOI | MR | Zbl

[32] W. Guo, Yu. V. Lutzeko, A. N. Skiba, “On nonnilpotent groups with every two 3-maximal subgroups permutable”, Siberian Math. J., 50 (2009), 988–997 | DOI | MR | Zbl

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

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

[35] X. Guo, K. P. Shum, “Cover-avoidance properties and the structure of finite groups”, Journal of Pure and Applied Algebra, 181 (2003), 297–308 | DOI | MR

[36] B. Li, A. N. Skiba, “New characterizations of finite supersoluble groups”, Science in China Series A: Mathematics, 50 (2008), 827–841 | MR

[37] Sh. Li, “Finite non-nilpotent groups all of whose second maximal subgroups are $TI$-groups”, Mathematical Proccedings of the Royal Irish Academy, 100 (2000), 65–71 | MR

[38] V. N. Kniahina, V. S. Monakhov, “On the permutability of $n$-maximal subgroups with Schmidt subgroups”, Trudy Inst. Mat. i Mekh. UrO RAN, 18, no. 3, 2012, 125–130

[39] V. S. Monakhov, V. N. Kniahina, “Finite groups with $\mathbb{P}$-subnormal subgroups”, Ricerche di Matematica, 62:2 (2013), 307–322 | DOI | MR | Zbl

[40] Yu. V. Lutsenko, A. N. Skiba, “Finite groups with subnormal second and third maximal subgroups”, Math. Notes, 91:5 (2012), 680–688 | DOI | MR | Zbl

[41] V. A. Kovaleva, A. N. Skiba, “Finite solvable groups with all $n$-maximal subgroups $\mathfrak{U}$-subnormal”, Sib. Math. J., 54 (2013), 65–73 | DOI | MR

[42] V. A. Kovaleva, A. N. Skiba, “Finite soluble groups with all $n$-maximal subgroups $\mathfrak{F}$-subnormal”, J. Group Theory, 17 (2014), 273–290 | DOI | MR | Zbl

[43] W. Guo, D. P. Andreeva, A. N. Skiba, “Finite groups of Spencer height $\leqslant3$”, Algebra Colloquium, 22 (2015), 437–444 | DOI | MR | Zbl

[44] V. A. Kovaleva, “Finite groups with generalized $\mathbb{P}$-subnormal second maximal subgroups”, Asian-European Journal of Mathematics, 7 (2014), 1450047-1–1450047-8 | DOI | MR

[45] V. A. Kovaleva, X. Yi, “Finite biprimary groups with all 3-maximal subgroups $\mathfrak{U}$-subnormal”, Acta Matematica Hungarica, 146 (2015), 47–55 | DOI | MR

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

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

[48] R. K. Agrawal, “Generalized center and hypercenter of a finite group”, Proc. Amer. Math. Soc., 54 (1976), 13–21 | DOI | MR

[49] L. A. Shemetkov, A. N. Skiba, Formations of Algebraic Systems, Nauka, M., 1989 | MR | Zbl

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