Abstract
We study the influence of weakly subnormal and partially subnormal subgroups on the structure of a group \( G \). In particular, we prove that a finite group \( G \) is supersoluble if and only if \( G=AB \), where \( A \) and \( B \) are supersoluble weakly subnormal subgroups in \( G \), and every Schmidt subgroup in \( G \) is partially subnormal in \( G \).
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References
Guo W., Shum K. P., and Skiba A. N., “\( X \)-Semipermutable subgroups of finite groups,” J. Algebra, vol. 315, no. 1, 31–41 (2007).
Hu B., Huang J., and Skiba A. N., “Finite groups with only \( {\mathfrak{F}} \)-normal and \( {\mathfrak{F}} \)-abnormal subgroups,” J. Group Theory, vol. 22, no. 5, 915–926 (2019).
Shemetkov L. A., Formations of Finite Groups [Russian], Nauka, Moscow (1978).
Doerk K. and Hawkes T., Finite Soluble Groups, De Gruyter, Berlin and New York (1992).
Baer R., “Classes of finite groups and their properties,” Illinois J. Math, vol. 1, no. 2, 115–187 (1957).
Between Nilpotent and Solvable, Weinstein M. (ed.), Polygonal Publ. House, Passaic (1982).
Vasilev A. F. and Vasileva T. I., “On finite groups whose principal factors are simple groups,” Russian Math. (Iz. VUZ), vol. 41, no. 11, 8–12 (1997).
Knyagina V. N. and Monakhov V. S., “Finite groups with subnormal Schmidt subgroups,” Sib. Math. J., vol. 45, no. 6, 1075–1079 (2004).
Monakhov V. S. and Chirik I. K., “On the supersoluble residual of a product of subnormal supersoluble subgroups,” Sib. Math. J., vol. 58, no. 2, 271–280 (2017).
Monakhov V. S., “On the supersolvable residual of mutually permutable products,” Probl. Fiz. Math. Tekh., vol. 1, no. 34, 69–70 (2018).
Monakhov V. S. and Trofimuk A. A., “Finite groups with two supersoluble subgroups,” J. Group Theory, vol. 22, no. 2, 297–312 (2019).
Asaad M. and Shaalan A., “On the supersolvability of finite groups,” Arch. Math., vol. 53, 318–326 (1989).
Semenchuk V. N., “Finite groups with a system of minimal non-\( \mathfrak{F} \)-subgroups,” in: Subgroup Structure of Finite Groups [Russian], Nauka i Tekhnika, Minsk (1981), 138–149.
Schmidt R., Subgroup Lattices of Groups, De Gruyter, Berlin (1994).
Bliznets I. V. and Selkin V. M., “On finite groups with modular Schmidt subgroup,” Probl. Fiz. Math. Tekh., vol. 4, no. 41, 36–38 (2019).
Monakhov V. S., “Product of a supersoluble and cyclic or primary group,” in: Finite Groups, Nauka i Tekhnika, Minsk (1978), 50–63.
Ballester-Bolinches A., Esteban-Romero R., and Asaad M., Products of Finite Groups, De Gruyter, Berlin and New York (2010).
Huppert B., Endliche Gruppen I, Springer, Berlin, Heidelberg, and New York (1967).
Chi Z. and Skiba A. N., “On a lattice characterization of finite soluble \( PST \)-groups,” Bull. Aust. Math. Soc., vol. 101, no. 2, 247–254 (2020). doi 10.1017/S0004972719000741
Vedernikov V. A., “Finite groups with subnormal Schmidt subgroups,” Algebra and Logic, vol. 46, no. 6, 363–372 (2007).
Skiba A. N., “On \( \sigma \)-subnormal and \( \sigma \)-permutable subgroups of finite groups,” J. Algebra, vol. 436, 1–16 (2015).
Beidleman J. C. and Skiba A. N., “On \( \tau_{\sigma} \)-quasinormal subgroups of finite groups,” J. Group Theory, vol. 20, no. 5, 955–964 (2017).
Al-Sharo K. A. and Skiba A. N., “On finite groups with \( \sigma \)-subnormal Schmidt subgroups,” Comm. Algebra, vol. 45, no. 10, 4158–4165 (2017).
Skiba A. N., “Some characterizations of finite \( \sigma \)-soluble \( P\sigma T \)-groups,” J. Algebra, vol. 495, no. 1, 114–129 (2018).
Skiba A. N., “On some classes of sublattices of the subgroup lattice,” J. Belarusian State Univ. Math. Informatics., vol. 3, 35–47 (2019).
Guo W. and Skiba A. N., “Finite groups whose \( n \)-maximal subgroups are \( \sigma \)-subnormal,” Sci. China Math., vol. 62, no. 7, 1355–1372 (2019).
Skiba A. N., “On sublattices of the subgroup lattice defined by formation Fitting sets,” J. Algebra, vol. 550, 69–85 (2020).
Ballester-Bolinches A., Kamornikov S. F., Pedraza-Aguilera M. C., and Pérez-Calabuig V., “On \( \sigma \)-subnormality criteria in finite \( \sigma \)-soluble groups,” RACSAM, vol. 114, no. 94 (2020). doi 10.1007/s13398-020-00824-4
Ballester-Bolinches A., Kamornikov S. F., Pedraza-Aguilera M. C., and Yi X., “On \( \sigma \)-subnormal subgroups of factorised finite groups,” J. Algebra, vol. 559, 195–202 (2020).
Yi X. and Kamornikov S. F., “Finite groups with \( \sigma \)-subnormal Schmidt subgroups,” J. Algebra, vol. 560, 181–191 (2020).
Kamornikov S. F. and Tyutyanov V. N., “On \( \sigma \)-subnormal subgroups of finite groups,” Sib. Math. J., vol. 60, no. 2, 266–270 (2020).
Kamornikov S. F. and Tyutyanov V. N., “On \( \sigma \)-subnormal subgroups of finite \( 3^{\prime} \)-groups,” Ukrainian Math. J., vol. 72, no. 6, 806–811 (2020).
Acknowledgment
The authors express their deep gratitude to the reviewer for useful comments and suggestions.
Funding
The authors were supported by the NNSF of China (Grant 11401264) and a TAPP of Jiangsu Higher Education Institutions (Grant PPZY 2015A013).
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Huang, J., Hu, B. & Skiba, A.N. Finite Groups with Weakly Subnormal and Partially Subnormal Subgroups. Sib Math J 62, 169–177 (2021). https://doi.org/10.1134/S0037446621010183
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DOI: https://doi.org/10.1134/S0037446621010183