Geodesic
On some aspects of the Kegel-Wielandt $\sigma$-problem
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2022), pp. 18-28.

Voir la notice de l'article provenant de la source Math-Net.Ru

For a partition $\sigma$ of the set $\mathbb{P}$ of all primes, a sufficient criterion for $\sigma$-subnormality of a subgroup in finite group is given. It is proved, that Kegel-Wielandt $\sigma$-problem has a positive solution in the class of all finite groups, in which all non-abelian composition factors are either alternating groups, or Suzuki groups, or sporadic groups.
Keywords: finite group, Hall subgroup, $\sigma$-subnormal subgroup, subnormal subgroup
Mots-clés : sporadic group. 
@article{IVM_2022_2_a1,
     author = {S. F. Kamornikov and V. N. Tyutyanov},
     title = {On some aspects of the {Kegel-Wielandt} $\sigma$-problem},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {18--28},
     publisher = {mathdoc},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2022_2_a1/}
}
TY  - JOUR
AU  - S. F. Kamornikov
AU  - V. N. Tyutyanov
TI  - On some aspects of the Kegel-Wielandt $\sigma$-problem
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2022
SP  - 18
EP  - 28
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2022_2_a1/
LA  - ru
ID  - IVM_2022_2_a1
ER  - 
%0 Journal Article
%A S. F. Kamornikov
%A V. N. Tyutyanov
%T On some aspects of the Kegel-Wielandt $\sigma$-problem
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2022
%P 18-28
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2022_2_a1/
%G ru
%F IVM_2022_2_a1
S. F. Kamornikov; V. N. Tyutyanov. On some aspects of the Kegel-Wielandt $\sigma$-problem. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2022), pp. 18-28. http://geodesic.mathdoc.fr/item/IVM_2022_2_a1/

[1] Kegel O. H., “Sylow-Gruppen und Subnormalteiler endlicher Gruppen”, Math. Z., 78 (1962), 205–221 | MR | Zbl

[2] Wielandt H., “Zusammengesetzte Gruppen: Hölders Programm heute”, Proc. Pure Math., 37 (1980), 161–173 | MR | Zbl

[3] Kleidman P. B., “A proof of the Kegel-Wielandt conjecture on subnormal subgroups”, Ann. Math., 133 (1991), 369–428 | MR | Zbl

[4] Kamornikov S. F., Selkin M. V., Podgruppovye funktory i klassy konechnykh grupp, Belarusk. navuka, Minsk, 2003

[5] Skiba A. N., “On $\sigma$-subnormal and $\sigma$-permutable subgroups of finite groups”, J. Algebra, 436 (2015), 1–16 | MR | Zbl

[6] The Kourovka notebook. Unsolved problems in group theory, Sobolev Inst. Math., Siberian Branch Russ. Acad. of Sci., Novosibirsk, 2018

[7] Kamornikov S. F., Tyutyanov V. N., “O $\sigma$-subnormalnykh podgruppakh konechnykh grupp”, Sib. matem. zhurn., 61:2 (2020), 337–343 | MR | Zbl

[8] Kamornikov S. F., Tyutyanov V. N., “O $\sigma$-subnormalnykh podgruppakh konechnykh $3^{'}$-grupp”, Ukr. matem. zhurn., 72:6 (2020), 806–811 | MR | Zbl

[9] Conway J. H., Curtis R. T., Norton S. P., Parker R. A., Wilson R. A., Atlas of finite groups, Oxford Univ. Press, Oxford, 1985 | MR | Zbl

[10] Wilson R. A., Maximal subgroups of sporadic groups, 19 Jan 2017, arXiv: 1701.02095v2 [math.GR] | DOI | MR

[11] Kamornikov S. F., Tyutyanov V. N., “Ob odnom kriterii $\sigma$-subnormalnosti podgruppy v konechnoi $3^{'}$-gruppe”, Izv. vuzov. Matem., 2020, no. 8, 36–43 | Zbl

[12] Doerk K., Hawkes T., Finite soluble groups, Walter de Gruyter, New York – Berlin, 1992 | MR

[13] Skiba A. N., “On some results in the theory of finite partially soluble groups”, Commun. Math. Stat., 4:3 (2016), 281–309 | MR | Zbl

[14] Skiba A. N., “On sublattices of the subgroup lattice defined by formation Fitting sets”, J. Algebra, 550:5 (2020), 69–85 | MR | Zbl

[15] Guralnik R., Kleidman P. B., Lyons R., “Sylow $p$-subgroups and subnormal subgroups of finite groups”, Proc. London Math. Soc., 66:3 (1993), 129–151 | MR

[16] Revin D. O., Vdovin E. P., “On the number of classes of conjugate Hall subgroups in finite simple groups”, J. Algebra, 324:12 (2010), 3614–3652 | MR | Zbl

[17] Vdovin E. P., Revin D. O., “Teoremy silovskogo tipa”, Usp. matem. nauk, 66:5 (2011), 3–46 | MR | Zbl

[18] Huppert B., Lempken W., “Simple groups of order divisible by at most four primes”, Izv. GGU im. F. Skoriny, Voprosy algebry, 3:16 (2000), 64–75 | Zbl