To the theorem of K. Doerk
Abstract
For a finite group $G$ and its maximal subgroup $M$ we proved that the generalized Fitting height of $G$ minus the generalized Fitting height of $M$ is not greater than 2 and the non-$p$-soluble length of $G$ minus the non-$p$-soluble length of $M$ is not greater than 1. We constructed a hereditary saturated formation $\mathfrak{F}$ such that $\{n_\sigma(G, \mathfrak{F})-n_\sigma(M, \mathfrak{F})\mid G$ is finite $\sigma$-soluble and $M$ is a maximal subgroup of $G\}=\mathbb{N}\cup\{0\}$ where $n_\sigma(G, \mathfrak{F})$ denotes the $\sigma$-nilpotent length of the $\mathfrak{F}$-residual of $G$.
This construction shows the results about the generalized lengths of maximal subgroups published in Math. Nachr. (1994) and Mathematics (2020) are not correct.
Keywords
finite group,
the generalized Fitting subgroup,
the generalized Fitting height,
the non-$p$-soluble length,
hereditary Plotkin radical,
$\sigma$-nilpotent group
Supporting agencies: The work was supported by BRFFR grant no. Φ23PHΦ-237.
About the Authors
V. I. Murashka
Francisk Skorina Gomel State University
Russian Federation
Russian Federation
Gomel
A. F. Vasil’ev
Francisk Skorina Gomel State University
Russian Federation
Russian Federation
Gomel
References
1. Cannon J. J., Eick B., Leedham-Green C. R. Special polycyclic generating sequences for finite soluble groups. J. Symb. Comput., 2004, vol. 38, iss. 5, pp. 1445–1460.
2. Khukhro E. I., Shumyatsky P. Nonsoluble and non-p-soluble length of finite groups. Isr. J. Math., 2015, vol. 207, iss. 2, pp. 507–525.
3. Khukhro E. I., Shumyatsky P. On the length of finite factorized groups. Ann. Mat. Pura Appl., 2015, vol. 194, iss. 6, pp. 1775–1780.
4. Fumagalli F., Leinen F., Puglisi O. A reduction theorem for nonsolvable finite groups. Isr. J. Math., 2019, vol. 232, iss. 1, pp. 231–260.
5. Guralnick R. M., Tracey G. On the generalized Fitting height and insoluble length of finite groups. Bull. London Math. Soc., 2020, vol. 52, iss. 5, pp. 924–931.
6. Khukhro E. I., Shumyatsky P. On the length of finite groups and of fixed points. Proc. Amer. Math. Soc., 2015, vol. 143, iss. 9, pp. 3781–3790.
7. Murashka V. I., Vasil’ev A. F. On the lengths of mutually permutable products of finite groups. Acta Math. Hungar., 2023, vol. 170, iss. 1, pp. 412–429.
8. Doerk K. U¨ ber die nilpotente La¨nge maximaler Untergruppen bei endlichen auflo¨sbaren Gruppen. Rend. Sem. Mat. Univ. Padova, 1994, vol. 91, pp. 20–21.
9. Monakhov V. S., Shpyrko O. A. The nilpotent π-length of maximum subgroups in finite π-soluble groups. Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2009, iss. 6, pp. 3–8.
10. Heliel A., Al-Shomrani M., Ballester-Bolinches A. On the σ-Length of Maximal Subgroups of Finite σ-Soluble Groups. Mathematics, 2020, vol. 8, iss. 12.
11. Plotkin B. I. Radicals in groups, operations on group classes and radical classes. Selected Questions of Algebra and Logic. Novosibirsk, Nauka, 1973, pp. 205–244 (in Russian).
12. Gardner B. J., Wiegandt R. Radical Theory of Rings. Marcel Dekker, New York, 2003.
13. Baer R. Group theoretical properties and functions. Colloq. Math., 1966, vol. 14, pp. 285–327.
14. Doerk K., Hawkes T. O. Finite Soluble Groups. De Gruyter Exp. Math., De Gruyter, Berlin, New York, 1992, vol. 4.
15. Skiba A. N. On σ-subnormal and σ-permutable subgroups of finite groups. J. Algebra, 2015, iss. 436, pp. 1–16.
16. Krempa J., Malinowska I. A. On Kurosh-Amitsur radicals of finite groups. An. Stiint. Univ. “Ovidius” Constanta Ser. Mat., 2011, vol. 19, iss. 1, pp. 175–190.
17. Ballester-Bolinches A., Pe´rez-Ramos M. D. A Note on the F-length of Maximal Subgroups in Finite Soluble Groups. Math. Nachr., 1994, vol. 166, iss. 1, pp. 67–70.
18. Huppert B., Blackburn N. Finite Groups III. Grundlehren Math. Wiss., Springer-Verlag, Berlin, Heidelberg, 1982, vol. 243.
Review
For citations:
Murashka V.I., Vasil’ev A.F. To the theorem of K. Doerk. Proceedings of the Institute of Mathematics of the NAS of Belarus. 2025;33(1):28-33.
Views:
12
Email this article 