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Licensed Unlicensed Requires Authentication Published by De Gruyter November 17, 2023

Finite groups with modular 𝜎-subnormal subgroups

  • A-Ming Liu , Mingzhu Chen , Inna N. Safonova ORCID logo and Alexander N. Skiba EMAIL logo
From the journal Journal of Group Theory

Abstract

Let 𝜎 be a partition of the set of prime numbers. In this paper, we describe the finite groups for which every 𝜎-subnormal subgroup is modular.

Award Identifier / Grant number: 12171126

Award Identifier / Grant number: 12101165

Award Identifier / Grant number: 12101166

Award Identifier / Grant number: 20211328

Award Identifier / Grant number: 20211778

Funding statement: Research was supported by the National Natural Science Foundation of China (No. 12171126, 12101165, and 12101166). Research of the third and the fourth authors were supported by Ministry of Education of the Republic of Belarus (No. 20211328 and 20211778).

Acknowledgements

The authors are deeply grateful for the useful comments and suggestions of the reviewers.

  1. Communicated by: Andrea Lucchini

References

[1] A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of Finite Groups, De Gruyter Exp. Math. 53, Walter de Gruyter, Berlin, 2010. 10.1515/9783110220612Search in Google Scholar

[2] A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite Groups, Math. Appl. (Springer) 584, Springer, Dordrecht, 2006. Search in Google Scholar

[3] A. Ballester-Bolinches, M. C. Pedraza-Aguilera and V. Pérez-Calabuig, On two classes of generalised finite T-groups, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 117 (2023), no. 3, Paper No. 105. 10.1007/s13398-023-01443-5Search in Google Scholar

[4] K. Doerk and T. Hawkes, Finite Soluble Groups, De Gruyter Exp. Math. 4, Walter de Gruyter, Berlin, 1992. 10.1515/9783110870138Search in Google Scholar

[5] W. Guo, Z. Chi and A. N. Skiba, On 𝜎-supersoluble groups and one generalization of C L T -groups, J. Algebra 512 (2018), 92–108. 10.1016/j.jalgebra.2018.07.008Search in Google Scholar

[6] B. Hu, J. Huang and A. N. Skiba, A generalisation of finite P T -groups, Bull. Aust. Math. Soc. 97 (2018), no. 3, 396–405. 10.1017/S0004972717001083Search in Google Scholar

[7] B. Hu, J. Huang and A. N. Skiba, On 𝜎-quasinormal subgroups of finite groups, Bull. Aust. Math. Soc. 99 (2019), no. 3, 413–420. 10.1017/S0004972718001132Search in Google Scholar

[8] B. Hu, J. Huang and A. N. Skiba, On two open problems of the theory of permutable subgroups of finite groups, Publ. Math. Debrecen 94 (2019), no. 3–4, 477–491. 10.5486/PMD.2019.8473Search in Google Scholar

[9] B. Huppert, Endliche Gruppen. I, Grundlehren Math. Wiss. 134, Springer, Berlin, 1967. 10.1007/978-3-642-64981-3Search in Google Scholar

[10] B. Huppert and N. Blackburn, Finite Groups. III, Grundlehren Math. Wiss. 243, Springer, Berlin, 1982. 10.1007/978-3-642-67997-1Search in Google Scholar

[11] N. Itô and J. Szép, Über die Quasinormalteiler von endlichen Gruppen, Acta Sci. Math. (Szeged) 23 (1962), 168–170. Search in Google Scholar

[12] R. Maier and P. Schmid, The embedding of quasinormal subgroups in finite groups, Math. Z. 131 (1973), 269–272. 10.1007/BF01187244Search in Google Scholar

[13] O. Ore, Contributions to the theory of groups of finite order, Duke Math. J. 5 (1939), no. 2, 431–460. 10.1215/S0012-7094-39-00537-5Search in Google Scholar

[14] D. J. S. Robinson, The structure of finite groups in which permutability is a transitive relation, J. Aust. Math. Soc. 70 (2001), no. 2, 143–159. 10.1017/S1446788700002573Search in Google Scholar

[15] I. N. Safonova and A. N. Skiba, One application of the theory 𝜎-soluble P σ T -groups, preprint (2022). Search in Google Scholar

[16] R. Schmidt, Subgroup Lattices of Groups, De Gruyter Exp. Math. 14, Walter de Gruyter, Berlin, 1994. 10.1515/9783110868647Search in Google Scholar

[17] A. N. Skiba, On 𝜎-subnormal and 𝜎-permutable subgroups of finite groups, J. Algebra 436 (2015), 1–16. 10.1016/j.jalgebra.2015.04.010Search in Google Scholar

[18] A. N. Skiba, On some results in the theory of finite partially soluble groups, Commun. Math. Stat. 4 (2016), no. 3, 281–309. 10.1007/s40304-016-0088-zSearch in Google Scholar

[19] A. N. Skiba, A Robinson characterization of finite P σ T -groups, preprint (2017), https://arxiv.org/abs/1709.06423. Search in Google Scholar

[20] A. N. Skiba, Some characterizations of finite 𝜎-soluble P σ T -groups, J. Algebra 495 (2018), 114–129. 10.1016/j.jalgebra.2017.11.009Search in Google Scholar

[21] J. G. Thompson, An example of core-free quasinormal subgroups of 𝑝-groups, Math. Z. 96 (1967), 226–227. 10.1007/BF01124081Search in Google Scholar

[22] G. Zacher, I gruppi risolubili finiti in cui i sottogruppi di composizione coincidono con i sottogruppi quasi-normali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 37 (1964), 150–154. Search in Google Scholar

Received: 2023-04-26
Revised: 2023-09-23
Published Online: 2023-11-17
Published in Print: 2024-05-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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