Geodesic
Characterizations of $\sigma$-Solvable Finite Groups
Matematičeskie zametki, Tome 111 (2022) no. 4, pp. 506-518.

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All the groups considered in this paper are finite, and $G$ always denotes a finite group; $\sigma$ is a partition of the set $\mathbb{P}$ of all primes, i.e., $\sigma=\{\sigma_{i} \mid i \in I\}$, where $\mathbb{P}=\bigcup_{i \in I} \sigma_{i}$ and $\sigma_{i} \cap \sigma_{j}=\varnothing$ for all $i \ne j$. A group $G$ is said to be $\sigma$-primary if $G$ is a $\sigma_{i}$-group for some $i=i(G)$, and $\sigma$-solvable if every chief factor of $G$ is $\sigma$-primary. A set of subgroups $\mathcal{H}$ of a group $G$ is called a complete Hall $\sigma$-set of $G$ if every element $\ne 1$ of the set $\mathcal{H}$ is a Hall $\sigma_{i}$-subgroup $G$ for some $i$, and $\mathcal{H}$ contains exactly one Hall $\sigma_{i}$-subgroup of the group $G$ for all $i$ such that $\sigma_{i}\cap \pi(G)\ne \varnothing$. A subgroup $A$ of a group $G$ is said to be $K$-$\mathfrak{S}_{\sigma}$-subnormal in $G$ if $G$ contains a series of subgroups $A=A_{0} \le A_{1} \le\cdots\le A_{t}=G$ such that either $A_{i-1} \trianglelefteq A_{i}$ or the group $A_{i}/(A_{i-1})_{A_{i}}$ is $\sigma$-solvable for all $i=1,\dots,t$. We say that a subgroup $A$ of a group $G$ is weakly $K$-$\mathfrak{S}_{\sigma}$-subnormal in $G$ if $G$ contains $K$-$\mathfrak{S}_{\sigma}$-subnormal subgroups $S$ and $T$ such that $G=AT$ and $A \cap T \le S \le A$. In the present paper, we study conditions under which a group is $\sigma$-solvable. In particular, we prove that a group $G$ is $\sigma$-solvable if and only if at least one of the following two conditions is satisfied: (i) $G$ has a complete Hall $\sigma$-set $\mathcal H$ all of whose elements are weakly $K$-$\mathfrak{S}_{\sigma}$-subnormal in $G$; (ii) in every maximal chain of subgroups $\cdots M_{3} M_{2} M_{1} M_{0}=G$ of the groups $G$, at least one of the subgroups $M_{3}$, $M_{2}$, or $M_{1}$ is weakly $K$-$\mathfrak{S}_{\sigma}$-subnormal in $G$.
Keywords: finite group, groups of equal order, $\sigma$-solvable group, $K$-$\mathfrak{S}_{\sigma}$-subnormal subgroup, weakly $K$-$\mathfrak{S}_{\sigma}$-subnormal subgroup. 
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W. Guo; Ch. Wan; I. N. Safonova; A. N. Skiba. Characterizations of $\sigma$-Solvable Finite Groups. Matematičeskie zametki, Tome 111 (2022) no. 4, pp. 506-518. http://geodesic.mathdoc.fr/item/MZM_2022_111_4_a2/

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

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

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

[4] A. N. Skiba, “Some characterizations of finite $\sigma$-soluble $P\sigma T$-groups”, J. Algebra, 495 (2018), 114–129 | DOI | MR | Zbl

[5] W. Guo, A. N. Skiba, “On $\sigma$-supersoluble groups and one generalization of $CLT$-groups”, J. Algebra, 512 (2018), 92–108 | DOI | MR | Zbl

[6] A. N. Skiba, “On sublattices of the subgroup lattice defined by formation Fitting sets”, J. Algebra, 550 (2020), 69–85 | DOI | MR | Zbl

[7] C. Zhang, Z. Wu, W. Guo, “On weakly $\sigma$-permutable subgroups of finite groups”, Publ. Math. Debrecen, 91:3–4 (2017), 489–502 | DOI | MR | Zbl

[8] J. Huang, B. Hu, A. N. Skiba, “On weakly $s$-quasinormal subgroups of finite groups”, Publ. Math. Debrecen, 92:1–2 (2018), 201–216 | MR | Zbl

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

[10] A. Ballester-Bolinches, J. C. Beidleman, H. Heineken, “Groups in which Sylow subgroups and subnormal subgroups permute”, Illinois J. Math., 47:1–2 (2003), 63–69 | MR | Zbl

[11] Y. Wang, “$c$-Normality of groups and its properties”, J. Algebra, 180 (1996), 954–965 | DOI | MR | Zbl

[12] R. Schmidt, Subgroup Lattices of Groups, Walter de Gruyter, Berlin, 1994 | MR | Zbl

[13] I. Zimmermann, “Submodular subgroups in finite group”, Math. Z., 202 (1989), 545–557 | DOI | MR | Zbl

[14] I. Zimmermann, “On a theorem of Deskins”, Proc. Amer. Math. Soc., 107:4 (1989), 895–899 | DOI | MR | Zbl

[15] V. A. Kovaleva, “A criterion for a finite group to be $\sigma$-soluble”, Comm. Algebra, 46:12 (2018), 5410–5415 | DOI | MR | Zbl

[16] K. A Al-Sharo, A. N. Skiba, “On finite groups with $\sigma$-subnormal Schmidt subgroups”, Comm. Algebra, 45 (2017), 4158–4165 | DOI | MR | Zbl

[17] J. C. Beidleman, A. N. Skiba, “On $\tau_{\sigma}$-quasinormal subgroups of finite groups”, J. Group Theory, 20:5 (2017), 955–969 | DOI | MR | Zbl

[18] J. Huang, B. Hu, X. Wu, “Finite groups all of whose subgroups are $\sigma$-subnormal or $\sigma$-abnormal”, Comm. Algebra, 45:1 (2017), 4542–4549 | DOI | MR | Zbl

[19] W. Guo, A. N. Skiba, “Finite groups whose $n$-maximal subgroups are $\sigma$-subnormal”, Sci. China. Math., 62 (2019), 1355–1372 | DOI | MR | Zbl

[20] Abd El-Rahman Heliel, M. Al-Shomrani, A. Ballester-Bolinches, “On the $\sigma$-length of maximal subgroups of finite $\sigma$-soluble groups”, Mathematics, 8:12 (2020), 2165 | MR

[21] A. Ballester-Bolinches, S. F. Kamornikov, M. C. Pedraza-Aguilera, V. Perez-Calabuig, “On $\sigma$-subnormality criteria in finite $\sigma$-soluble groups”, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 114 (2020), 94 | MR | Zbl

[22] S. F. Kamornikov, V. N. Tyutyanov, “O $\sigma$-subnormalnykh podgruppakh konechnykh grupp”, Sib. matem. zhurn., 61:2 (2020), 337–343 | DOI | Zbl

[23] X. Yi, S. F. Kamornikov, “Finite groups with $\sigma$-subnormal Schmidt subgroups”, J. Algebra, 560:15 (2020), 181–191 | MR | Zbl

[24] S. F. Kamornikov, V. N. Tyutyanov, “On $\sigma$-subnormal subgroups of finite $3'$-groups”, Ukrainian Math. J., 72:6 (2020), 935–941 | DOI | MR | Zbl

[25] M. M. Al-Shomrani, A. A. Heliel, A. Ballester-Bolinches, “On $\sigma$-subnormal closure”, Comm. Algebra, 48:8 (2020), 3624–3627 | DOI | MR | Zbl

[26] A-Ming Liu, W. Guo, I. N. Safonova, A. N. Skiba, “$G$-covering subgroup systems for some classes of $\sigma$-soluble groups”, J. Algebra, 585 (2021), 280–293 | DOI | MR | Zbl

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

[28] R. Schmid, “Endliche Gruppen mit vielen modularen Untergruppen”, Abhan. Math. Sem. Univ. Hamburg., 34 (1969), 115–125 | DOI | MR | Zbl

[29] J. Lu, W. Meng, “Finite groups with non-subnormal subgroups”, Comm. Algebra, 45:5 (2017), 2043–2046 | DOI | MR | Zbl

[30] W. Guo, Structure Theory for Canonical Classes of Finite Groups, Springer, Heidelberg, 2015 | MR | Zbl

[31] K. Doerk, T. Hawkes, Finite Soluble Group, Walter de Gruyter, Berlin, 1992 | MR

[32] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967 | MR | Zbl