On groups with formational subnormal strictly 2-maximal subgroups

Abstract

UDC 512.542

Let $H$ be a subgroup of a finite group $G.$
If $G$ contains a maximal subgroup $M$ such that $H$ is a maximal subgroup in $M,$ then $H$ is called a $2$-maximal subgroup of $G.$
A subgroup $U$ of $G$ is said to be a strictly $2$-maximal subgroup in $G$ if $U$ is a $2$-maximal subgroup of $G$ and $U$ is not a 2-maximal subgroup in any proper subgroup of $G.$
We investigate the finite groups with $\mathfrak X$-subnormal strictly $2$-maximal subgroups for arbitrary subgroup-closed formation $\mathfrak X.$
In such a group, any proper subgroup has a nilpotent $\mathfrak X$-residual.
We study in more detail the case where $\mathfrak X= \mathfrak A_1\mathfrak F$ for a subgroup-closed formation $\mathfrak F$ and the case where $\mathfrak X$ is a soluble saturated formation.

 

References

V. S. Monakhov, Vvedenie v teoriyu konechny`kh grupp i ikh klassov, Vy`she`jsh. shk., Minsk (2006).

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

V. S. Monakhov, V. N. Kniahina, Finite groups with $Bbb{P}$-subnormal subgroups<.em>, Ric. Mat., 62, № 1, 307 – 322 (2013), https://doi.org/10.1007/s11587-013-0153-9 DOI: https://doi.org/10.1007/s11587-013-0153-9

M. Asaad, Finite groups some of whose $n$-maximal subgroups are normal, Acta Math. Hungar., 54, № 1-2, 9 – 27 (1989), https://doi.org/10.1007/BF01950705 DOI: https://doi.org/10.1007/BF01950705

B. Huppert, Normalteiler und maximale Untergruppen endlicher Gruppen, Math. Z., 60, 409 – 434 (1954), https://doi.org/10.1007/BF01187387 DOI: https://doi.org/10.1007/BF01187387

Yu. V. Luczenko, A. N. Skiba, Konechny`e gruppy` s subnormal`ny`mi vtory`mi ili tret`imi maksimal`ny`mi podgruppami, Mat. zametki, 91, № 5, 730 – 740 (2012).

V. A. Kovaleva, A. N. Skiba, Finite soluble groups with all n-maximal subgroups ${germ{F}}$-subnormal, J. Group Theory, 17, № 1, 273 – 290 (2014), https://doi.org/10.1515/jgt-2013-0047 DOI: https://doi.org/10.1515/jgt-2013-0047

V. S. Monakhov, O gruppakh s formaczionno subnormal`ny`mi 2-maksimal`ny`mi podgruppami, Mat. zametki, 105, № 2, 69 – 277 (2019).

V. S. Monakhov, Konechny`e gruppy` s abnormal`ny`mi i U-subnormal`ny`mi podgruppami, Sib. mat. zhurn., 57, № 2, 447 – 462 (2016).

A. F. Vasil`ev, T. I. Vasil`eva, V. N. Tyutyanov,O konechny`kh gruppakh sverkhrazreshimogo tipa, Sib. mat. zhurn., 51, № 6, 1270 – 1281 (2010).

V. I.Murashko, Svojstva klassa konechny`kh grupp s P-subnormal`ny`mi cziklicheskimi primarny`mi podgruppami, Dokl. NAN Belarusi, 58, № 1, 5 – 8 (2014).

A. Ballester-Bolinches, L. M. Ezquerro, Classes of finite groups, Springer-Verlag, Dordrecht (2006).

L. A. Shemetkov, Formaczii konechny`kh grupp, Nauka, Moskva (1978).

Yu. V. Gorbatova, M. N. Konovalova, Konechny`e gruppy` s subnormal`ny`mi strogo 2-maksimal`ny`mi ili strogo 3-maksimal`ny`mi podgruppami, Vestn. Omsk. un-ta, 24, № 3, 4 – 12 (2019).

Published
22.01.2021
How to Cite
Monakhov, V. S., and M. N. Konovalova. “On Groups With Formational Subnormal Strictly 2-Maximal Subgroups”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 1, Jan. 2021, pp. 107 -16, doi:10.37863/umzh.v73i1.1115.
Section
Research articles