Abstract
In this paper, we study the structure of finite groups in which some Sylow subgroup is \(\mathcal M\)-permutable. In particular, we mainly reveal the structure of its formation and the properties of \(p\)-modular subgroups of some quotient groups.
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References
K. Doerk and T. O. Hawkes, Finite Soluble Groups (De Gruyter, Berlin, 1992).
B. Huppert and N. Blackburn, Finite Groups. III (Springer-Verlag, Berlin–New York, 1982).
B. Huppert, Endliche Gruppen. I (Springer-Verlag, Berlin–New York, 1967).
P. Hall, “A characteristic property of solvable groups,” J. Lond. Math. Soc. 12, 188–200 (1937).
J. Tate, “Nilpotent quotient groups,” J. Algebra 3 (suppl.1), 109–111 (1964).
L. Miao and J. Tang, “A new condition for solvable groups,” J. Pure Appl. Algebra 221 (10), 2504–2510 (2017).
L. Miao and J. Zhang, “On a class of nonsolvable groups,” J. Algebra 496, 1–10 (2018).
W. Burnside, Theory of Groups of Finite Order (Cambridge University Press, Cambridge, 1911).
P. Hall, “A contribution to the theory of groups of prime-power order,” Proc. London Math. Soc. 36 (2), 29–95 (1934).
P. Apisa and B. Klopsch, “A generalization of the Burnside basis theorem,” J. Algebra 400, 8–16 (2014).
L. Miao and W. Lempken, “On \(\mathcal M\)-supplemented subgroups of finite groups,” J. Group Theory 12 (2), 271–287 (2009).
L. Miao and W. Lempken, “On \(\mathcal M\)-permutable Sylow subgroups of finite groups,” Illinois J. Math. 53 (4), 1095–1107 (2009).
H. Bao and L. Miao, “Finite groups with some \(\mathcal M\)-permutable primary subgroups,” Bull. Malays. Math. Sci. Soc. 36, 1041–1048 (2013).
L. Miao and W. Lempken, “On \(\mathcal M\)-permutable maximal subgroups of Sylow subgroups of finite groups,” Comm. Algebra 38 (10), 3649–3659 (2010).
P. Hall, “Theorems like Sylow’s,” Proc. London Math. Soc. (3) 6, 286–304 (1956).
W. Guo, The Theory of Classes of Groups (Science Press, Beijing/New York, 2000).
W. Guo, K. P. Shum, and A. N. Skiba, “Criterions of supersolubility for products of supersoluble groups,” Publ. Math. Debrecen 68 (3–4), 433–449 (2006).
V. S. Monakhov and V. N. Kniahina, “Finite groups with complemented subgroups of prime orders,” J. Group Theory 18 (6), 905–912 (2015).
Acknowledgments
The authors wish to thank the support of NSFC (projects no. 11871062 and 12161062), NSFC-RFBR (project no. 12011530061), and the Natural Science Foundation of Jiangsu Province (project no. BK20181451).
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Wang, Y., Liu, W., Miao, L. et al. The Structure of Finite Groups with an \(\mathcal M\)-Permutable Sylow Subgroup. Math Notes 113, 129–137 (2023). https://doi.org/10.1134/S0001434623010133
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DOI: https://doi.org/10.1134/S0001434623010133