Abstract
Let \(\sigma =\{\sigma_i |i\in I\}\) be some partition of the set of all primes \(\mathbb{P}\) and G be a finite group. A group is said to be \(\sigma\)-primary if it is a finite \(\sigma_{i}\)-group for some i. A subgroup A of G is said to be \({\sigma}\)-subnormal in G if there is a subgroup chain \(A=A_{0} \leq A_{1} \leq \cdots \leq A_{t}=G\) such that either \(A_{i-1}\trianglelefteq A_{i}\) or \(A_{i}/(A_{i-1})_{A_{i}}\) is \(\sigma\)-primary for all \(i=1, \ldots , t\). A subgroup S of G is m- \(\sigma\)-permutable in G if \(S=\langle M, B \rangle\) for some modular subgroup M and \(\sigma\)-permutable subgroup B of G. We say that a subgroup H of G is m-\(\sigma\)-embedded in G if there exist an m-\(\sigma\)-permutable subgroup S and a \(\sigma\)-subnormal subgroup T of G such that \(H^G=HT\) and \(H\cap T\leq S\leq H\), where \(H^G = \langle H^x | x \in G \rangle\) is the normal closure of H in G.
In this paper, we study the properties of m-\(\sigma\)-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized.
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References
A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of Finite Groups, Walter de Gruyter (Berlin–New York, 2010).
K. Doerk and T. Hawkes, Finite Soluble Groups, Walter de Gruyter (Berlin, 1992).
W. Guo, Structure Theory for Canonical Classes of Finite Groups, Springer (Heidelberg–New York–Dordrecht–London, 2015).
W. Guo and A. N. Skiba, Finite groups with given s-embedded and n-embedded subgroups, J. Algebra, 321(2009), 2843-2860.
B. Huppert, Endliche Gruppen. I, Springer-Verlag (Berlin–Heidelberg–New York, 1967).
B. N. Knyagina and V. S. Monakhov, On \(\pi'\)-properties of finite groups having a Hall \(\pi\)-subgroup, Siberian Math. J., 522 (2011), 398–309.
R. Schmidt, Subgroup Lattices of Groups, Walter de Gruyter (Berlin, 1994).
L. A. Shemetkov and A. N. Skiba, On the \(\mathfrak{X}\Phi\)-hypercentre of finite groups, J. Algebra, 322 (2009), 2106–2117.
A. N. Skiba, A generalization of a Hall theorem, J. Algebra Appl., 15 (2015), 21–36.
A. N. Skiba, On \(\sigma\)-subnormal and \(\sigma\)-permutable subgroups of finite groups, J. Algebra, 436 (2015), 1–16.
A. N. Skiba, On some results in the theory of finite partially soluble groups, Comm. Math. Stat., 4 (2016), 281–309.
Y. Wang, c-normality of groups and its properties, J. Algebra, 180 (1996), 954–965.
X. Wei, On m-\(\sigma\)-permutable and weakly m-\(\sigma\)-permutable subgroups of finite groups, Comm. Algebra, 47 (2019), 945–956.
M. Weinstein, Between Nilpotent and Solvable, Polygonal Publishing House (Passaic, 1982).
Z. Wu, C. Zhang and J. Huang, Finite groups with given \(\sigma\)-embedded and n-\(\sigma\)-embedded subgroups, Indian J. Pure Appl. Math., 48 (2017), 429–448.
C. Zhang and A. N. Skiba, On two sublattices of the subgroup lattice of a finite group, J. Group Theory, 22 (2019), 1035–1047.
C. Zhang and A. N. Skiba, On a lattice characterisation of finite soluble PST-groups, Bull. Aust. Math. Soc., 101 (2020), 247–254.
C. Zhang, Z. Wu and W. Guo, On weakly \(\sigma\)-permutable subgroups of finite groups, Publ. Math. Debrecen, 91 (2017), 489–502.
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Research supported by the NSFC of China (No. 12001526, 11961017, 12026238, 12026212) and Natural Science Foundation of Jiangsu Province, China (No. BK20200626)
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Guo, J., Guo, W., Qiao, S. et al. On m-\(\sigma\)-embedded subgroups of finite groups. Acta Math. Hungar. 165, 100–111 (2021). https://doi.org/10.1007/s10474-021-01176-0
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DOI: https://doi.org/10.1007/s10474-021-01176-0