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On $\sigma $-permutably embedded subgroups of finite groups
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 11-24 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $\sigma =\{\sigma _i\colon i\in I\}$ be some partition of the set of all primes $\mathbb {P}$, $G$ be a finite group and $\sigma (G)=\{\sigma _i\colon \sigma _i\cap \pi (G)\neq \emptyset \}$. A set $\mathcal {H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every non-identity member of $\mathcal {H}$ is a Hall $\sigma _i$-subgroup of $G$ and $\mathcal {H}$ contains exactly one Hall $\sigma _i$-subgroup of $G$ for every $\sigma _i\in \sigma (G)$. $G$ is said to be $\sigma $-full if $G$ possesses a complete Hall $\sigma $-set. A subgroup $H$ of $G$ is $\sigma $-permutable in $G$ if $G$ possesses a complete Hall $\sigma $-set $\mathcal {H}$ such that $HA^x$= $A^xH$ for all $A\in \mathcal {H}$ and all $x\in G$. A subgroup $H$ of $G$ is $\sigma $-permutably embedded in $G$ if $H$ is $\sigma $-full and for every $\sigma _i\in \sigma (H)$, every Hall $\sigma _i$-subgroup of $H$ is also a Hall $\sigma _i$-subgroup of some $\sigma $-permutable subgroup of $G$. \endgraf By using the $\sigma $-permutably embedded subgroups, we establish some new criteria for a group $G$ to be soluble and supersoluble, and also give the conditions under which a normal subgroup of $G$ is hypercyclically embedded. Some known results are generalized.
Let $\sigma =\{\sigma _i\colon i\in I\}$ be some partition of the set of all primes $\mathbb {P}$, $G$ be a finite group and $\sigma (G)=\{\sigma _i\colon \sigma _i\cap \pi (G)\neq \emptyset \}$. A set $\mathcal {H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every non-identity member of $\mathcal {H}$ is a Hall $\sigma _i$-subgroup of $G$ and $\mathcal {H}$ contains exactly one Hall $\sigma _i$-subgroup of $G$ for every $\sigma _i\in \sigma (G)$. $G$ is said to be $\sigma $-full if $G$ possesses a complete Hall $\sigma $-set. A subgroup $H$ of $G$ is $\sigma $-permutable in $G$ if $G$ possesses a complete Hall $\sigma $-set $\mathcal {H}$ such that $HA^x$= $A^xH$ for all $A\in \mathcal {H}$ and all $x\in G$. A subgroup $H$ of $G$ is $\sigma $-permutably embedded in $G$ if $H$ is $\sigma $-full and for every $\sigma _i\in \sigma (H)$, every Hall $\sigma _i$-subgroup of $H$ is also a Hall $\sigma _i$-subgroup of some $\sigma $-permutable subgroup of $G$. \endgraf By using the $\sigma $-permutably embedded subgroups, we establish some new criteria for a group $G$ to be soluble and supersoluble, and also give the conditions under which a normal subgroup of $G$ is hypercyclically embedded. Some known results are generalized.
DOI : 10.21136/CMJ.2018.0148-17
Classification : 20D10, 20D20, 20D35
Keywords: finite group; $\sigma $-subnormal subgroup; $\sigma $-permutably embedded subgroup; \hbox {$\sigma $-soluble} group; supersoluble group 
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Cao, Chenchen; Zhang, Li; Guo, Wenbin. On $\sigma $-permutably embedded subgroups of finite groups. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 11-24. doi: 10.21136/CMJ.2018.0148-17

[1] Asaad, M.: On the solvability of finite groups. Arch. Math. 51 (1988), 289-293. | DOI | MR | JFM

[2] Asaad, M.: On maximal subgroups of Sylow subgroups of finite groups. Commun. Algebra 26 (1998), 3647-3652. | DOI | MR | JFM

[3] Asaad, M., Heliel, A. A.: On $S$-quasinormally embedded subgroups of finite groups. J. Pure Appl. Algebra 165 (2001), 129-135. | DOI | MR | JFM

[4] Asaad, M., Ramadan, M., Shaalan, A.: Influence of $\pi$-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group. Arch. Math. 56 (1991), 521-527. | DOI | MR | JFM

[5] Ballester-Bolinches, A.: Permutably embedded subgroups of finite soluble groups. Arch. Math. 65 (1995), 1-7. | DOI | MR | JFM

[6] Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of Finite Groups. De Gruyter Expositions in Mathematics 53, Walter de Gruyter, Berlin (2010). | MR | JFM

[7] Ballester-Bolinches, A., Pedraza-Aguilera, M. C.: On minimal subgroups of finite groups. Acta Math. Hung. 73 (1996), 335-342. | DOI | MR | JFM

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

[9] Bray, H. G., Deskins, W. E., Johnson, D., Humphreys, J. F., Puttaswamaiah, B. M., Venzke, P., Walls, G. L.: Between Nilpotent and Solvable. Polygonal Publ. House, Washington (1982). | MR | JFM

[10] Buckley, J. T.: Finite groups whose minimal subgroups are normal. Math. Z. 116 (1970), 15-17. | DOI | MR | JFM

[11] Chen, X., Guo, W., Skiba, A. N.: Some conditions under which a finite group belongs to a Baer-local formation. Commun. Algebra 42 (2014), 4188-4203. | DOI | MR | JFM

[12] Doerk, K., Hawkes, T.: Finite Soluble Groups. De Gruyter Expositions in Mathematics 4, Walter de Gruyter, Berlin (1992). | MR | JFM

[13] Gorenstein, D.: Finite Groups. Harper's Series in Modern Mathematics, Harper and Row, Publishers, New York (1968). | MR | JFM

[14] Guo, W.: The Theory of Classes of Groups. Mathematics and Its Applications 505, Kluwer Academic Publishers, Dordrecht; Science Press, Beijing (2000). | DOI | MR | JFM

[15] Guo, W.: Structure Theory for Canonical Classes of Finite Groups. Springer, Berlin (2015). | DOI | MR | JFM

[16] Guo, W., Cao, C., Skiba, A. N., Sinitsa, D. A.: Finite groups with $\mathcal{H}$-permutable subgroups. Commun. Math. Stat. 5 (2017), 83-92. | DOI | MR | JFM

[17] Guo, W., Skiba, A. N.: Finite groups with generalized Ore supplement conditions for primary subgroups. J. Algebra 432 (2015), 205-227. | DOI | MR | JFM

[18] Guo, W., Skiba, A. N.: Finite groups with permutable complete Wielandt sets of subgroups. J. Group Theory 18 (2015), 191-200. | DOI | MR | JFM

[19] Guo, W., Skiba, A. N.: Groups with maximal subgroups of Sylow subgroups $\sigma$-permutably embedded. J. Group Theory 20 (2017), 169-183. | DOI | MR | JFM

[20] Guo, W., Skiba, A. N.: On $\Pi$-quasinormal subgroups of finite groups. Monatsh. Math. 185 (2018), 443-453. | DOI | MR | JFM

[21] Huppert, B.: Endliche Gruppen I. Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen 134. Springer, Berlin German (1967). | DOI | MR | JFM

[22] Huppert, B., Blackburn, N.: Finite groups III. Grundlehren der Mathematischen Wissenschaften 243, Springer, Berlin (1982). | DOI | MR | JFM

[23] Li, B.: On $\Pi$-property and $\Pi$-normality of subgroups of finite groups. J. Algebra 334 (2011), 321-337. | DOI | MR | JFM

[24] Li, Y., Wang, Y.: On $\pi$-quasinormally embedded subgroups of finite group. J. Algebra 281 (2004), 109-123. | DOI | MR | JFM

[25] Schmidt, R.: Subgroup Lattices of Groups. De Gruyter Expositions in Mathematics 14, Walter de Gruyter, Berlin (1994). | MR | JFM

[26] Skiba, A. N.: On weakly $s$-permutable subgroups of finite groups. J. Algebra 315 (2007), 192-209. | DOI | MR | JFM

[27] Skiba, A. N.: On two questions of L. A. Shemetkov concerning hypercyclically embedded subgroups of finite groups. J. Group Theory 13 (2010), 841-850. | DOI | MR | JFM

[28] Skiba, A. N.: A characterization of the hypercyclically embedded subgroups of finite groups. J. Pure Appl. Algebra 215 (2011), 257-261. | DOI | MR | JFM

[29] Skiba, A. N.: On $\sigma$-subnormal and $\sigma$-permutable subgroups of finite groups. J. Algebra 436 (2015), 1-16. | DOI | MR | JFM

[30] Skiba, A. N.: On some results in the theory of finite partially soluble groups. Commun. Math. Stat. 4 (2016), 281-309. | DOI | MR | JFM

[31] Srinivasan, S.: Two sufficient conditions for supersolvability of finite groups. Isr. J. Math. 35 (1980), 210-214. | DOI | MR | JFM

[32] Zhang, C., Wu, Z., Guo, W.: On weakly $\sigma$-permutable subgroups of finite groups. Pub. Math. Debrecen. 91 (2017), 489-502. | DOI | MR

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