On the modularity of the lattice of Baer-σ-local formations
https://doi.org/10.29235/1561-2430-2023-59-1-7-17
Abstract
Throughout this paper, all groups are finite. A group class closed under taking homomorphic images and finite subdirect products is called a formation. The symbol σ denotes some partition of the set of all primes. V. G. Safonov, I. N. Safonova, A. N. Skiba (Commun. Algebra. 2020. Vol. 48, № 9. P. 4002–4012) defined a generalized formation σ-function. Any function f of the form f : σ È {Ø} → {formations of groups}, where f(Ø) ≠ ∅, is called a generalized formation σ-function. Generally local formations or so-called Baer-σ-local formations are defined by means of generalized formation σ-functions. The set of all such formations partially ordered by set inclusion is a lattice. In this paper it is proved that the lattice of all Baerσ-local formations is algebraic and modular.
About the Author
N. N. Vorob’ev
Vitebsk State University named after P. M. Masherov
Belarus
Belarus
Nikolay N. Vorob’ev – Dr. Sc. (Physics and Mathematics), Associate Professor, Professor at the Department of Mathematics
33, Moscow Ave., 210038, Vitebsk
References
1. Shemetkov L. A. Formations of Finite Groups. Moscоw, Nauka Publ., 1978. 272 p. (in Russian).
2. Shemetkov L. A., Skiba A. N. Formations of Algebraic Systems. Moscоw, Nauka Publ., 1989. 256 p. (in Russian).
3. Skiba A. N. Algebra of Formations. Minsk, Belaruskaya navuka Publ., 1997. 240 p. (in Russian).
4. Skiba A. N., Shemetkov L. A. Multiply L-composite formations of finite groups. Ukrainian Mathematical Journal, 2000, vol. 52, no. 6, pp. 898–913. https://doi.org/10.1007/BF02591784
5. Zhang Chi, Skiba A. N. On -closed classes of finite groups. Ukrainian Mathematical Journal, 2018, vol. 70, no. 12, pp. 1966–1977. https://doi.org/10.1007/s11253-019-01619-6
6. Zhang Chi, Safonov V. G., Skiba A. N. On n-multiply σ-local formations of finite groups. Communications in Algebra, 2019, vol. 47, no. 3, pp. 957–968. https://doi.org/10.1080/00927872.2018.1498875
7. Safonov V. G., Safonova, I. N., Skiba A. N. On Baer-σ-local formations of finite groups. Communications in Algebra, 2020, vol. 48, no. 9, pp. 4002–4012. https://doi.org/10.1080/00927872.2020.1753760
8. Skiba A. N. On σ-subnormal and σ-permutable subgroups of finite groups. Journal of Algebra, 2015, vol. 436, pp. 1–16. https://doi.org/10.1016/j.jalgebra.2015.04.010
9. Safonov V. G., Safonova, I. N., Skiba A. N. On one generalization of σ-local and Baer-local formations. Problemy fiziki matematiki i tekhniki = Problems of Physics, Mathematics and Technics, 2019, no. 4 (41), pp. 65–69.
10. Skiba A. N. Some characterizations of finite σ-soluble PσT-groups. Journal of Algebra, 2018, vol. 495, pp. 114–129. https://doi.org/10.1016/j.jalgebra.2017.11.009
11. Skiba A. N. On sublattices of the subgroup lattice defined by formation Fitting sets. Journal of Algebra, 2020, vol. 550, pp. 69–85. https://doi.org/10.1016/j.jalgebra.2019.12.013
12. Artamonov V. A., Saliĩ V. N., Skornyakov L. A., Shevrin L. N., Shul’geǐfer E. G. General Algebra. Vol. 2. Moscоw, Nauka Publ., 1991. 480 p. (in Russian).
13. Skiba A. N. On local formations of length 5. Arifmeticheskoe i podgruppovoe stroenie konechnykh grupp: trudy Gomel’skogo seminara [Arithmetical and Subgroup Construction of Finite Groups. Works of Gomel Seminar]. Minsk, Nauka i tekhnika Publ., 1986, pp. 135–149 (in Russian).
14. Skiba A. N., Shemetkov L. A. Multiply w-local formations and Fitting classes of finite groups. Siberian Advances in Mathematics, 2000, vol. 10, no. 2, pp. 112–141.
15. Vorob’ev N. N. Algebra of Classes of Finite Groups. Vitebsk, Vitebsk University Press, 2012. 322 p. (in Russian).
16. Ballester-Bolinches A., Shemetkov L. A. On lattices of p-local formations of finite groups. Mathematische Nachrichten, 1997, vol. 186, pp. 57–65. https://doi.org/10.1002/mana.3211860103
17. Safonov V. G. On a question of the theory of totally saturated formations of finite groups. Algebra Colloquium, 2008, vol. 15, no. 1, pp. 119–128. https://doi.org/10.1142/S1005386708000126
18. Safonov V. G. On modularity of the lattice of totally saturated formations of finite groups. Communications in Algebra, 2007, vol. 35, no. 11, pp. 3495–3502. https://doi.org/10.1080/00927870701509354
19. Shabalina I. P. On the lattice of t-closed n-multiply w-local formations of finite groups. Vestsі Natsyyanalʼnai akademіі navuk Belarusі. Seryya fіzіka-matematychnykh navuk = Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series, 2003, no. 1, pp. 28–30 (in Russian).
20. Shcherbina V. V., Safonov V. G. On sublattices of the lattice of partially totally saturated formations of finite groups. Belgorod State University Scientific Bulletin Mathematics Physics, 2019, vol. 51, no. 1, pp. 66–89. https://doi.org/10.18413/2075-4639-2019-51-1-66-89
21. Shcherbina V. V., Safonov V. G. On some properties of the lattice of totally saturated formations of finite groups. Belgorod State University Scientific Bulletin Mathematics Physics, 2019, vol. 51, no. 2, pp. 227–244 (in Russian). https://doi.org/10.18413/2075-4639-2019-51-2-227-244
22. Shemetkov L. A., Skiba A. N., Vorob’ev N. N. On lattices of formations of finite groups. Algebra Colloquium, 2010, vol. 17, no. 4, pp. 557–564. https://doi.org/10.1142/S1005386710000532
23. Shemetkov L. A., Skiba A. N., Vorob’ev N. N. On laws of lattices of partially saturated formations. Asian-European Journal of Mathematics, 2009, vol. 02, no. 01, pp. 155–169. https://doi.org/10.1142/S1793557109000133
24. Tsarev A. A. On the lattice of all totally composition formations of finite groups. Ricerche di Matematica, 2019, vol. 68, no. 2, pp. 693–698. https://doi.org/10.1007/s11587-019-00433-3
25. Tsarev A. A., Vorob’ev N. N. Lattices of composition formations of finite groups and laws. Journal of Algebra and Its Applications, 2018, vol. 17, no. 05, pp. 1850084-1–1850084-17. https://doi.org/10.1142/S0219498818500846
26. Tsarev A. A., Vorob’ev N. N. On a question of the theory of partially composition formations. Algebra Colloquium, 2014, vol. 21, no. 3, pp. 437–447. https://doi.org/10.1142/S1005386714000388
27. Vorob’ev N. N., Skiba A. N., Tsarev A. A. Laws of the lattices of partially composition formations. Siberian Mathematical Journal, 2011, vol. 52, no. 5, pp. 802–812. https://doi.org/10.1134/S0037446611050059
28. Vorob’ev N. N., Tsarev A. A. On the modularity of the lattice of t-closed n-multiply w-composite formations. Ukrainian Mathematical Journal, 2010, vol. 62, no. 4, pp. 518–529. https://doi.org/10.1007/s11253-010-0368-9
29. Zhiznevsky P. A. On modularity and inductance of the lattice of all t-closed n-multiply w-composition formations of finite groups. Izvestiya Gomel’skogo gosudarstvennogo universiteta imeni F. Skoriny = Proceedings of the Francisk Scorina Gomel State University, 2010, no. 1 (58), pp. 185–191 (in Russian).
30. Tsarev A. A. Algebraic lattices of partially saturated formations of finite groups. Afrika Matematika, 2020, vol. 31, no. 3–4, pp. 701–707. https://doi.org/10.1007/s13370-019-00753-5
31. Safonov V. G. The property of being algebraic for the lattice of all t-closed totally saturated formations. Algebra and Logic, 2006, vol. 45, no. 5, pp. 353–356. https://doi.org/10.1007/s10469-006-0032-5
32. Shabalina I. P. Algebraic lattices of t-closed n-multiply w-local formations. Izvestiya Gomel’skogo gosudarstvennogo universiteta imeni F. Skoriny. Voprosy algebry – 18 = Proceedings of the Francisk Scorina Gomel State University. Algebra Questions – 18, 2002, no. 5 (14), pp. 59–67 (in Russian).
33. Shcherbina V. V. Algebraicity of lattice of t-closed totally w-saturated formations of finite groups. Ufimskii Matematicheskii Zhurnal, 2020, vol. 12, no. 1, pp. 82–90. https://doi.org/10.13108/2020-12-1-82
34. Tsarev A. A., Kukharev A. V. Algebraic lattices of solvably saturated formations and their applications. Boletìn de la Sociedad Matemática Mexicana, 2020, vol. 26, no. 3, pp. 1003–1014. https://doi.org/10.1007/s40590-020-00290-3
35. Shcherbina V. V. On two problems of the theory of partially totally composition formations of finite groups. Applied Mathematics & Physics, 2020, vol. 52, no. 1, pp. 18–32 (in Russian). https://doi.org/10.18413/2687-0959-2020-52-1-18-32
36. Vorob’ev N. N., Chachuyeu A. V. On the smallest definition of a Baer formation. Vesnik Vitsebskaga dzyarzhaunaga universiteta = Vestnik of Vitebsk State University, 2022, no. 3 (116), pp. 15–18 (in Russian).
37. Skiba A. N., Vorob’ev N. N. On the lattices of saturated and solubly saturated formations of finite groups. Southeast Asian Bulletin of Mathematics, 2013, vol. 37, no. 5, pp. 771–780.
Review
Views: 384
Email this article 