Geodesic
$\mathfrak F_\tau$-embedded and $\mathfrak F_{\tau,\Phi}$-embedded subgroups of finite groups
Algebra i logika, Tome 54 (2015) no. 3, pp. 351-380.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\mathfrak F$ be a nonempty formation of groups, $\tau$ a subgroup functor, and $H$ a $p$-subgroup of a finite group $G$. Suppose also that $\bar G=G/H_G$ and $\bar H=H/H_G$. We say that $H$ is $\mathfrak F_\tau$-embedded ($\mathfrak F_{\tau,\Phi}$-embedded) in $G$ if, for some quasinormal subgroup $\bar T$ of $\bar G$ and some $\tau$-subgroup $\bar S$ of $\bar G$ contained in $\bar H$, the subgroup $\bar H\bar T$ is $S$-quasinormal in $\bar G$ and $\bar H\cap\bar T\le\bar SZ_\mathfrak F(\bar G)$ (resp., $\bar H\cap\bar T\le\bar SZ_{\mathfrak F,\Phi}(\bar G)$). Using the notions of $\mathfrak F_\tau$-embedded and $\mathfrak F_{\tau,\Phi}$-embedded subgroups, we give some characterizations of the structure of finite groups. A number of earlier concepts and related results are further developed and unified.
Keywords: finite group, subgroup functor, $\mathfrak F_\tau$-embedded subgroup, $\mathfrak F_{\tau,\Phi}$-embedded subgroup, supersoluble group. 
@article{AL_2015_54_3_a3,
     author = {X. Chen and W. Guo and A. N. Skiba},
     title = {$\mathfrak F_\tau$-embedded and $\mathfrak F_{\tau,\Phi}$-embedded subgroups of finite groups},
     journal = {Algebra i logika},
     pages = {351--380},
     publisher = {mathdoc},
     volume = {54},
     number = {3},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2015_54_3_a3/}
}
TY  - JOUR
AU  - X. Chen
AU  - W. Guo
AU  - A. N. Skiba
TI  - $\mathfrak F_\tau$-embedded and $\mathfrak F_{\tau,\Phi}$-embedded subgroups of finite groups
JO  - Algebra i logika
PY  - 2015
SP  - 351
EP  - 380
VL  - 54
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2015_54_3_a3/
LA  - ru
ID  - AL_2015_54_3_a3
ER  - 
%0 Journal Article
%A X. Chen
%A W. Guo
%A A. N. Skiba
%T $\mathfrak F_\tau$-embedded and $\mathfrak F_{\tau,\Phi}$-embedded subgroups of finite groups
%J Algebra i logika
%D 2015
%P 351-380
%V 54
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2015_54_3_a3/
%G ru
%F AL_2015_54_3_a3
X. Chen; W. Guo; A. N. Skiba. $\mathfrak F_\tau$-embedded and $\mathfrak F_{\tau,\Phi}$-embedded subgroups of finite groups. Algebra i logika, Tome 54 (2015) no. 3, pp. 351-380. http://geodesic.mathdoc.fr/item/AL_2015_54_3_a3/

[1] A. Ballester-Bolinches, R. Esteban-Romero, M. Asaad, Products of finite groups, de Gruyter Exp. Math., 53, Walter de Gruyter, Berlin, 2010 | DOI | MR | Zbl

[2] A. Ballester-Bolinches, M. C. Pedraza-Aguilera, “Sufficient conditions for supersolubility of finite groups”, J. Pure Appl. Algebra, 127:2 (1998), 113–118 | DOI | MR | Zbl

[3] Y. Wang, “$C$-normality groups and its properties”, J. Algebra, 180:3 (1996), 954–965 | DOI | MR | Zbl

[4] W. Guo, A. N. Skiba, “Finite groups with given $s$-embedded and $n$-embedded subgroups”, J. Algebra, 321:10 (2009), 2843–2860 | DOI | MR | Zbl

[5] W. Guo, K. P. Shum, A. N. Skiba, “On solubility and supersolubility of some classes of finite groups”, Sci. China Ser. A, 52:2 (2009), 1–15 | DOI | MR

[6] I. A. Malinowska, “Finite groups with $sn$-embedded or $s$-embedded subgroups”, Acta Math. Hungar., 136:1/2 (2012), 76–89 | DOI | MR | Zbl

[7] J. Huang, “On $\mathfrak F_s$-quasinormal subgroups of finite groups”, Commun. Algebra, 38:11 (2010), 4063–4076 | DOI | MR | Zbl

[8] L. Miao, B. Li, “On $\mathfrak F$-quasinormal primary subgroups of finite groups”, Commun. Algebra, 39:10 (2011), 3515–3525 | DOI | MR | Zbl

[9] A. Ballester-Bolinches, L. M. Ezquerro, Classes of finite groups, Math. Appl. (Springer), 584, Springer-Verlag, Dordrecht, 2006 | MR | Zbl

[10] K. Doerk, T. Hawkes, Finite soluble groups, De Gruyter Expo. Math., 4, Walter de Gruyter, Berlin etc., 1992 | MR

[11] W. Guo, The theory of classes of groups, Math. Appl. (Dordrecht), 505, Kluwer Acad. Publ., Dordrecht; Sci. Press, Beijing, 2000 | MR | Zbl

[12] R. Maier, P. Schmid, “The embedding of quasinormal subgroups in finite groups”, Math. Z., 131:3 (1973), 269–272 | DOI | MR | Zbl

[13] O. Kegel, “Sylow–Gruppen und Subnormalteiler endlicher Gruppen”, Math. Z., 78:1 (1962), 205–221 | DOI | MR | Zbl

[14] P. Schmid, “Subgroups permutable with all Sylow subgroups”, J. Algebra, 207:1 (1998), 285–293 | DOI | MR | Zbl

[15] W. Guo, A. N. Skiba, “On factorizations of finite groups with $\mathfrak F$-hypercentral intersections of the factors”, J. Group Theory, 14:5 (2011), 695–708 | DOI | MR | Zbl

[16] W. Guo, “On $\mathfrak F$-supplemented subgroups of finite groups”, Manuscr. Math., 127:2 (2008), 139–150 | DOI | MR | Zbl

[17] L. A. Shemetkov, A. N. Skiba, “On the $\mathfrak X\Phi$-hypercentre of finite groups”, J. Algebra, 322:6 (2009), 2106–2117 | DOI | MR | Zbl

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

[19] A. N. Skiba, L. A. Shemetkov, “Multiply $\mathfrak L$-composition formations of finite groups”, Ukr. Math. J., 52:6 (2000), 898–913 | DOI | MR | Zbl

[20] D. Gorenstein, Finite groups, Harper's Series in Modern Mathematics, Harper Row Publ., New York a.o., 1968 | MR | Zbl

[21] B. Huppert, N. Blackburn, Finite groups, v. III, Grundlehren Math. Wiss., 243, Springer-Verlag, Berlin a.o., 1982 | MR | Zbl

[22] A. N. Skiba, “On two questions of L. A. Shemetkov concerning hypercyclically embedded subgroups of finite groups”, J. Group Theory, 13:6 (2010), 841–850 | DOI | MR | Zbl

[23] T. M. Gagen, Topics in finite groups, London Math. Soc. Lecture Note Ser., 16, Cambridge Univ. Press, Cambridge etc., 1976 | MR | Zbl

[24] L. Dornhoff, “$M$-groups and 2-groups”, Math. Z., 100:3 (1967), 226–256 | DOI | MR | Zbl

[25] H. N. Ward, “Automorphisms of quaternion-free 2-groups”, Math. Z., 112:1 (1969), 52–58 | DOI | MR | Zbl

[26] Y. G. Berkovich, L. S. Kazarin, “Indices of elements and normal structure of finite groups”, J. Algebra, 283:2 (2005), 564–583 | DOI | MR | Zbl

[27] A. Ballester-Bolinches, M. D. Pérez-Ramos, “On $\mathfrak F$-subnormal subgroups and Frattini-like subgroups of a finite group”, Glasg. Math. J., 36:2 (1994), 241–247 | DOI | MR | Zbl

[28] B. Huppert, Endliche Gruppen, v. I, Grundlehren mathem. Wiss. Einzeldarstel., 134, Springer-Verlag, Berlin a.o., 1967 | DOI | MR | Zbl

[29] V. N. Semechuk, “Minimalnye ne $\mathfrak F$-gruppy”, Algebra i logika, 18:3 (1979), 348–382 | MR

[30] B. Li, “On $\Pi$-property and $\Pi$-normality of subgroups of finite groups”, J. Algebra, 334:1 (2011), 321–337 | DOI | MR | Zbl

[31] L. M. Ezquerro, X. Soler-Escrivà, “Some permutability properties related to Fhypercentrally embedded subgroups of finite groups”, J. Algebra, 264:1 (2003), 279–295 | DOI | MR | Zbl

[32] P. Schmid, Subgroup lattices of groups, Walter de Gruyter, Berlin, 1994 | MR | Zbl

[33] A. Ballester-Bolinches, L. M. Ezquerro, “A note on the Jordan–Hölder theorem”, Rend. Semin. Mat. Univ. Padova, 80 (1987), 25–32 | MR

[34] S. Li, Z. Shen, J. Liu, X. Liu, “The influence of SS-quasinormality of some subgroups on the structure of finite groups”, J. Algebra, 319:10 (2008), 4275–4287 | DOI | MR | Zbl

[35] X. Chen, W. Guo, “On the partial $\Pi$-property of subgroups of finite groups”, J. Group Theory, 16:5 (2013), 745–766 | MR | Zbl

[36] Z. Chen, “On a theorem of Srinivasan”, J. Southwest Teach. Univ., Ser. B, 1987, no. 1, 1–4 (Chinese)

[37] Q. Zhang, L. Wang, “The influence of S-semipermutable subgroups on the structure of finite groups”, Acta Math. Sin., 48:1 (2005), 81–88 (Chinese) | MR | Zbl

[38] V. O. Lukyanenko, A. N. Skiba, “On weakly $\tau$-quasinormal subgroups of finite groups”, Acta Math. Hung., 125:3 (2009), 237–248 | DOI | MR | Zbl

[39] S. Chen, V. Go, “O slabo $S$-vlozhennykh i slabo $\tau$-vlozhennykh podgruppakh”, Sib. matem. zh., 54:5 (2013), 1162–1181 | MR | Zbl

[40] K. A. Al-Sharo, “On nearly $S$-permutable subgroups of finite groups”, Commun. Algebra, 40:1 (2012), 315–326 | DOI | MR | Zbl

[41] W. Guo, A. N. Skiba, “Finite groups with systems of $\Sigma$-embedded subgroups”, Sci. China Math., 54:9 (2011), 1909–1926 | DOI | MR | Zbl

[42] S. Li, J. Liu, “A generalization of cover-avoiding properties in finite groups”, Commun. Algebra, 39:4 (2011), 1455–1464 | DOI | MR | Zbl

[43] W. Guo, K. P. Shum, A. N. Skiba, “Conditionally permutable subgroups and supersolubility of finite groups”, Southeast Asian Bull. Math., 29:3 (2005), 493–510 | MR | Zbl

[44] B. Khu, V. Go, “$c$-poluperestanovochnye podgruppy konechnykh grupp”, Sib. matem. zh., 48:1 (2007), 224–235 | MR | Zbl