On $\sigma$-subnormal subgroups of finite 3'-groups

  • S. F. Kamornikov Francisk Skorina Gomel State University
  • V. N. Tyutyanov Gomel. Phil. Int. University "MITSO", Belarus

Abstract

UDC 512.542

For a partition $\sigma$ of the set $\mathbb{R}$ of all primes, it is solved that if every complete Hall set of type $\sigma$ of a finite $3 \prime$-group $G$ is reducible in some subgroup $H$ of $G$, then $H$ is $\sigma$ -subnormal in $G$.

 

References

O. H. Kegel, Sylow-Gruppen und Subnormalteiler endlicher Gruppen, Math. Z., 78, 205 – 221 (1962) https://doi.org/10.1007/BF01195169 DOI: https://doi.org/10.1007/BF01195169

H. Wielandt, Zusammengesetzte Gruppen: Holders Programm heute, Proc. Pure Math., 37, 161 – 173 (1980).

P. B. Kleidman, A proof of the Kegel – Wielandt conjecture on subnormal subgroups, Ann. Math., 133, 369 – 428 (1991) https://doi.org/10.2307/2944342 DOI: https://doi.org/10.2307/2944342

R. Guralnick, P. B. Kleidman, R. Lyons, Sylow $p$-subgroups and subnormal subgroups of nite groups, Proc. London Math. Soc., 66, No 3, 129 – 151 (1993) https://doi.org/10.1112/plms/s3-66.1.129 DOI: https://doi.org/10.1112/plms/s3-66.1.129

Нерешенные вопросы теории групп: Коуровская тетрадь (Russian) [[Nereshenny`e voprosy` teorii grupp: Kourovskaya tetrad`, ]], In-t matematiki SO RAN, Novosibirsk (2018).

A. N. Skiba, On some results in the theory of nite partially soluble groups, Commun. Math. Stat., 4, No 3, 281 – 309 (2016) https://doi.org/10.1007/s40304-016-0088-z DOI: https://doi.org/10.1007/s40304-016-0088-z

A. N. Skiba, On $sigma$ -subnormal and $sigma$-permutable subgroups of nite groups, J. Algebra, 436, 1 – 16 (2015) https://doi.org/10.1016/j.jalgebra.2015.04.010 DOI: https://doi.org/10.1016/j.jalgebra.2015.04.010

A. N. Skiba, О $sigma$ -свойствах конечных групп I (Russian) [[O $sigma$-svojstvakh konechny`kh grupp I ]] , Problemy` fiziki, matematiki i tekhniki, No 4(21), 89–96 (2014).

M. Suzuki, On a class double transitive groups, Ann. Math., 75, No 1, 105 – 145 (1962) https://doi.org/10.2307/1970423 DOI: https://doi.org/10.2307/1970423

K. Doerk, T. Hawkes, Finite soluble groups, Walter de Gruyter, Berlin; New York (1992) https://doi.org/10.1515/9783110870138 DOI: https://doi.org/10.1515/9783110870138

Published
17.06.2020
How to Cite
KamornikovS. F., and TyutyanovV. N. “On $\sigma$-Subnormal Subgroups of Finite 3’-Groups”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 6, June 2020, pp. 806-11, doi:10.37863/umzh.v72i6.1037.
Section
Research articles