Abstract
A subgroup A of a finite group G is said to be NS-supplemented in G, if there exists a subgroup B of G such that G=AB and whenever X is a normal subgroup of A and \(p\in \pi(B)\), there exists a Sylow p-subgroup Bp of B such that \(XB_p=B_pX\). In this paper, we prove the supersolubility of a group G in the following cases: every non-cyclic Sylow subgroup of G is NS-supplemented in G; G is soluble and all maximal subgroups of every non-cylic Sylow subgroup of G are NS-supplemented in G. The solubility of a group with NS-supplemented maximal subgroups is obtained.
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Arroyo-Jordá, M., Arroyo-Jordá, P., Martínez-Pastor, A., Pérez-Ramos, M.D.: On finite products of groups and supersolubility. J. Algebra 323, 2922–2934 (2010)
Asaad, M., Heliel, A.A.: On \(S\)-quasinormal embedded subgroups of finite groups. J. Pure App. Algebra 165, 129–135 (2001)
Asaad, M., Ramadan, M., Shaalan, A.: Influence of \(\pi \)-quasinormality on maximal subgroups of Sylow subgroups of fitting subgroup of a finite group. Arch. Math. 56, 521–527 (1991)
A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of Finite Groups, (Walter de Gruyter (Berlin–New York, 2010)
Ballester-Bolinches, A., Pedraza-Aguilera, M.C.: Sufficient conditions for supersolvability of finite groups. J. Pure App. Algebra 127, 113–118 (1998)
Guo, W.: Finite groups with seminormal Sylow subgroups. Acta Math. Sinica 24, 1751–1758 (2008)
Guo, W., Shum, K.P., Skiba, A.N.: Criterions of supersolubility for products of supersoluble groups. Publ. Math. Debrecen 68, 433–449 (2006)
Guralnick, R.M.: Subgroups of prime power index in a simple group. J. Algebra 81, 304–311 (1983)
B. Huppert, Endliche Gruppen. I, Springer (Berlin–Heidelberg–New York, 1967)
Li, S., Shen, Z., Liu, J., Liu, X.: The influence of SS-quasinormality of some subgroups on the structure of finite groups. J. Algebra 319, 4275–4287 (2008)
Monakhov, V.S.: Finite groups with seminormal Hall subgroups. Math. Notes 80, 542–549 (2006)
Monakhov, V.S., Chirik, I.K.: On the \(p\)-supersolvability of a finite factorizable group with normal factors. Trudy Inst. Mat. Mekh. UrO RAN 21, 256–267 (2015)
Monakhov, V.S., Trofimuk, A.A.: Finite groups with subnormal non-cyclic subgroups. J. Group Theory 17, 889–895 (2014)
Monakhov, V.S., Trofimuk, A.A.: Supersolubility of a finite group with normally embedded maximal subgroups in Sylow subgroups. Siberian Math. J. 59, 922–930 (2018)
Skiba, A.N.: On weakly \(s\)-permutable subgroups of finite groups. J. Algebra 315, 192–209 (2007)
Srinivasan, S.: Two sufficient conditions for supersolvability of finite groups. Israel J. Math. 35, 210–214 (1980)
The GAP Group: GAP — Groups, Algorithms, and Programming, Ver. GAP 4.9.2 released on 4 July 2018, http://www.gap-system.org
Tyutyanov, V.N., Kniahina, V.N.: Finite groups with biprimary Hall subgroups. J. Algebra 443, 430–440 (2015)
Walter, J.H.: Characterization of finite groups with abelian Sylow 2-subgroups. Ann. of Math. 89, 405–514 (1969)
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Monakhov, V.S., Trofimuk, A.A. On the supersolubility of a finite group with NS-supplemented subgroups. Acta Math. Hungar. 160, 161–167 (2020). https://doi.org/10.1007/s10474-019-00997-4
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DOI: https://doi.org/10.1007/s10474-019-00997-4