Finite groups with 𝑂𝑆-propermutable subgroups
https://doi.org/10.22405/2226-8383-2021-22-3-457-463
Abstract
A subgroup 𝐴 of a group 𝐺 is called 𝑂𝑆-propermutable in 𝐺 if there is a subgroup 𝐵 such that 𝐺 = 𝑁𝐺(𝐴)𝐵, 𝐴𝐵 is a subgroup of 𝐺 and the subgroup 𝐴 permutes with all Schmidt subgroups of 𝐵. In this situation, the subgroup 𝐵 is called 𝑂𝑆-prosupplement to 𝐴 in 𝐺.
In this paper, we proved the 𝑝-solubility of a finite group 𝐺 such that a Sylow 𝑝-subgroup of 𝐺 is 𝑂𝑆-propermutable in 𝐺, where 𝑝 > 5.
About the Author
Ekaterina Vladimirovna Zubei
Brest State A. S. Pushkin University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
References
1. Berkovich, Ya. G., Pal’chik, Je. M. 1967, “On the commutability of subgroups of afinite group“,
2. Sibirskii matematicheskii zhurnal, vol. 8, no. 4, pp. 741–753.
3. Kniahina, V. N., Monakhov, V. S. 2007, “Finite groups with seminormal Schmidt subgroups“,
4. Algebra i logika, vol. 46, no. 4, pp. 448–458.
5. Knyagina, V. N., Monakhov, V. S. 2011, “On the permutability of maximal subgroups with
6. Schmidt subgroups“, Trudy Instituta matematiki i mekhaniki Ural’skogo otdelenija Rossijskoj
7. akademii nauk, vol. 17, no. 4, pp. 126–133.
8. Knyagina, V. N., Monakhov, V. S. 2010, “On permutability of Sylow subgroups with Schmidt
9. subgroups“, Trudy Instituta matematiki i mekhaniki Ural’skogo otdelenija Rossijskoj akademii
10. nauk, vol. 16, no. 3, pp. 130–139.
11. Knyagina, V. N., Monakhov, V. S. 2012, “On the permutability of 𝑛-maximal subgroups with
12. Schmidt subgroups“, Trudy Instituta matematiki i mekhaniki Ural’skogo otdelenija Rossijskoj
13. akademii nauk, vol. 18, no. 3, pp. 125–130.
14. Knyagina, V. N., Monakhov, V. S. 2011, “On the 𝜋′-properties of a finite group possessing a
15. Hall 𝜋-subgroup“, Siberian Math. J., vol. 52, no. 2, pp. 234–243
16. Monakhov, V. S. 2006, “Vvedenie v teoriju konechnyh grupp i ih klassov“, Vyshjejshaja shkola,
17. Minsk.
18. Monakhov, V. S. “The Schmidt subgroups, its existence, and some of their classes“, Trudy
19. Ukrainskogo matematicheskogo kongressa: sbornik trudov. Kiev, 2002, pp. 81–90.
20. Monakhov, V. S., Zubei, E. V. 2018 “On composition factors of а finite group with 𝑂𝑆-seminormal
21. sylow subgroup“, Proceedings of the Institute of Mathematics of the National Academy
22. of Sciences of the Republic of Belarus, vol. 26, no. 1, pp. 90-94.
23. Monakhov, V. S., Trofimuk, A. A. 2020, “On the supersolvability of a group with seminormal
24. subgroups“, Sibirskii matematicheskii zhurnal, vol. 61, no. 1, pp. 148-159.
25. Shmidt, O. Ju. 1924 “Groups, whose all subgroups are special“, Matematicheskii sbornik, vol.
26. , no. 3–4, pp. 366–372.
27. Huppert, B. 1967 “Endliche Gruppen I“, Berlin, Heidelberg, New York.
28. Yi, X., Skiba, A. N. 2015 “On 𝑆-propermutable subgroups of finite groups“, Bull. Malays. Math.
29. Sci. Soc., vol. 38, no. 2, pp. 605-616.
Review
For citations:
Zubei E.V. Finite groups with 𝑂𝑆-propermutable subgroups. Chebyshevskii Sbornik. 2021;22(3):457-463. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-3-457-463
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