Abstract
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 \(G\).
Similar content being viewed by others
References
O. H. Kegel, “Untergruppenverbande endlicher Gruppen, die den subnormalteilerverband echt enthalten,” Arch. Math. 30 (3), 225–228 (1978).
A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite Groups (Springer, Dordrecht, 2006).
A. N. Skiba, “On \(\sigma\)-subnormal and \(\sigma\)-permutable subgroups of finite groups,” J. Algebra 436, 1–16 (2015).
A. N. Skiba, “Some characterizations of finite \(\sigma\)-soluble \(P\sigma T\)-groups,” J. Algebra 495, 114–129 (2018).
W. Guo and A. N. Skiba, “On \(\sigma\)-supersoluble groups and one generalization of \(CLT\)-groups,” J. Algebra 512, 92–108 (2018).
A. N. Skiba, “On sublattices of the subgroup lattice defined by formation Fitting sets,” J. Algebra 550, 69–85 (2020).
C. Zhang, Z. Wu, and W. Guo, “On weakly \(\sigma\)-permutable subgroups of finite groups,” Publ. Math. Debrecen 91 (3–4), 489–502 (2017).
J. Huang, B. Hu, and A. N. Skiba, “On weakly \(s\)-quasinormal subgroups of finite groups,” Publ. Math. Debrecen 92 (1–2), 201–216 (2018).
A. Ballester-Bolinches, R. Esteban-Romero, and M. Asaad, Products of Finite Groups (Walter de Gruyter, Berlin, 2010).
A. Ballester-Bolinches, J. C. Beidleman, and H. Heineken, “Groups in which Sylow subgroups and subnormal subgroups permute,” Illinois J. Math. 47 (1–2), 63–69 (2003).
Y. Wang, “\(c\)-Normality of groups and its properties,” J. Algebra 180, 954–965 (1996).
R. Schmidt, Subgroup Lattices of Groups (Walter de Gruyter, Berlin, 1994).
I. Zimmermann, “Submodular subgroups in finite group,” Math. Z. 202, 545–557 (1989).
I. Zimmermann, “On a theorem of Deskins,” Proc. Amer. Math. Soc. 107 (4), 895–899 (1989).
V. A. Kovaleva, “A criterion for a finite group to be \(\sigma\)-soluble,” Comm. Algebra 46 (12), 5410–5415 (2018).
K. A Al-Sharo and A. N. Skiba, “On finite groups with \(\sigma\)-subnormal Schmidt subgroups,” Comm. Algebra 45, 4158–4165 (2017).
J. C. Beidleman and A. N. Skiba, “On \(\tau_{\sigma}\)-quasinormal subgroups of finite groups,” J. Group Theory 20 (5), 955–969 (2017).
J. Huang, B. Hu, and X. Wu, “Finite groups all of whose subgroups are \(\sigma\)-subnormal or \(\sigma\)-abnormal,” Comm. Algebra 45 (1), 4542–4549 (2017).
W. Guo and A. N. Skiba, “Finite groups whose \(n\)-maximal subgroups are \(\sigma\)-subnormal,” Sci. China. Math. 62, 1355–1372 (2019).
Abd El-Rahman Heliel, M. Al-Shomrani, and A. Ballester-Bolinches, “On the \(\sigma\)-length of maximal subgroups of finite \(\sigma\)-soluble groups,” Mathematics 8 (12), 2165 (2020).
A. Ballester-Bolinches, S. F. Kamornikov, M. C. Pedraza-Aguilera, and V. Perez-Calabuig, “On \(\sigma\)- subnormality criteria in finite \(\sigma\)-soluble groups,” Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 114, 94 (2020).
S. F. Kamornikov and V. N. Tyutyanov, “On \(\sigma\)-subnormal subgroups of finite groups,” Sib. Math. J. 61 (2), 266–270 (2020).
X. Yi and S. F. Kamornikov, “Finite groups with \(\sigma\)-subnormal Schmidt subgroups,” J. Algebra 560 (15), 181–191 (2020).
S. F. Kamornikov and V. N. Tyutyanov, “On \(\sigma\)-subnormal subgroups of finite \(3'\)-groups,” Ukrainian Math. J. 72 (6), 935–941 (2020).
M. M. Al-Shomrani, A. A. Heliel, and A. Ballester-Bolinches, “On \(\sigma\)-subnormal closure,” Comm. Algebra 48 (8), 3624–3627 (2020).
A-Ming Liu, W. Guo, I. N. Safonova, and A. N. Skiba, “\(G\)-covering subgroup systems for some classes of \(\sigma\)-soluble groups,” J. Algebra 585, 280–293 (2021).
A. E. Spencer, “Maximal nonnormal chains in finite groups,” Pacific J. Math. 27, 167–173 (1968).
R. Schmid, “Endliche Gruppen mit vielen modularen Untergruppen,” Abhan. Math. Sem. Univ. Hamburg. 34, 115–125 (1969).
J. Lu and W. Meng, “Finite groups with non-subnormal subgroups,” Comm. Algebra 45 (5), 2043–2046 (2017).
W. Guo, Structure Theory for Canonical Classes of Finite Groups (Springer, Heidelberg, 2015).
K. Doerk and T. Hawkes, Finite Soluble Groups (Walter de Gruyter, Berlin, 1992).
B. Huppert, Endliche Gruppen. I (Springer- Verlag, Berlin, 1967).
Acknowledgments
The authors are deeply grateful to the referee for useful remarks and suggestions.
Funding
The research was supported by grants of the National Natural Science Foundation of China (grants no. 12171126 and 12101165). The work of the third author was supported by the Ministry of Education of the Republic of Belarus (under the project 20211328). The work the fourth author was supported by the Belarusian Republican Foundation for Fundamental Research (grant F20R-291).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2022, Vol. 111, pp. 506-518 https://doi.org/10.4213/mzm13301.
Rights and permissions
About this article
Cite this article
Guo, W., Wang, Z., Safonova, I.N. et al. Characterizations of \(\sigma\)-Solvable Finite Groups. Math Notes 111, 534–543 (2022). https://doi.org/10.1134/S000143462203021X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143462203021X