Geodesic
On strictly $2$-maximal subgroups of finite groups
Problemy fiziki, matematiki i tehniki, no. 4 (2021), pp. 95-100.

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

We give examples of finite soluble and simple groups in which every $2$-maximal subgroup is strictly $2$-maximal. We prove that if in a group $G$ there is a strictly $2$-maximal subgroup of order $2$, then $G$ is a supersoluble group of order $2pq$, where $p$ and $q$ are primes, not necessarily distinct, or $G$ is isomorphic to the alternating group $A_4$. We establish the structure of a finite group in which every $2$-maximal subgroup is a Hall subgroup. We prove that the requirement of $\mathfrak{F}$-subnormality of all strictly $2$-maximal subgroups coincides with the requirement of subnormality of all $2$-maximal subgroups of a group $G$ for a subgroup-closed saturated lattice formation $\mathfrak{F}$ containing all nilpotent groups and $G\notin\mathfrak{F}$.
Keywords: finite group, $2$-maximal subgroup, strictly $2$-maximal subgroup, Hall subgroup, lattice formation. 
@article{PFMT_2021_4_a15,
     author = {M. N. Konovalova and V. S. Monakhov and I. L. Sokhor},
     title = {On strictly $2$-maximal subgroups of finite groups},
     journal = {Problemy fiziki, matematiki i tehniki},
     pages = {95--100},
     publisher = {mathdoc},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PFMT_2021_4_a15/}
}
TY  - JOUR
AU  - M. N. Konovalova
AU  - V. S. Monakhov
AU  - I. L. Sokhor
TI  - On strictly $2$-maximal subgroups of finite groups
JO  - Problemy fiziki, matematiki i tehniki
PY  - 2021
SP  - 95
EP  - 100
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PFMT_2021_4_a15/
LA  - ru
ID  - PFMT_2021_4_a15
ER  - 
%0 Journal Article
%A M. N. Konovalova
%A V. S. Monakhov
%A I. L. Sokhor
%T On strictly $2$-maximal subgroups of finite groups
%J Problemy fiziki, matematiki i tehniki
%D 2021
%P 95-100
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PFMT_2021_4_a15/
%G ru
%F PFMT_2021_4_a15
M. N. Konovalova; V. S. Monakhov; I. L. Sokhor. On strictly $2$-maximal subgroups of finite groups. Problemy fiziki, matematiki i tehniki, no. 4 (2021), pp. 95-100. http://geodesic.mathdoc.fr/item/PFMT_2021_4_a15/

[1] GAP — Groups, Algorithms, Programming: a System for Computational Discrete Algebra, Ver. 4.11.1 released on 02 March 2021, (Date of access: 20.04.2021) http://www.gapsystem.org

[2] V.S. Monakhov, V.N. Kniahina, “Finite groups with $\mathbb{P}$-subnormal subgroups”, Ricerche Mat., 62 (2013), 307–322 | DOI | MR | Zbl

[3] H. Meng, X. Guo, “Weak second maximal subgroups in solvable groups”, J. Algebra, 517 (2019), 112–118 | DOI | MR | Zbl

[4] Kourovskaya tetrad. Nereshennye voprosy teorii grupp, 19-e izd. dop., Institut matematiki Sibirskogo otdeleniya RAN, Novosibirsk, 2018, 248 pp.

[5] H. Meng, X. Guo, “Overgroups of weak second maximal subgroups”, Bull. Aust. Math. Soc., 99 (2019), 83–88 | DOI | MR | Zbl

[6] Ts. Chzhan, Ch. Gao, L. Myao, “O vtorykh maksimalnykh podgruppakh konechnykh grupp”, Sibirskii matematicheskii zhurnal, 62 (2021), 221–225 | Zbl

[7] N.V. Maslova, D.O. Revin, “Konechnye gruppy, vse maksimalnye podgruppy kotorykh khollovy”, Matematicheskie trudy, 22:2 (2012), 105–126

[8] V.S. Monakhov, “Konechnye $\pi$-razreshimye gruppy s khollovymi maksimalnymi podgruppami”, Matematicheskie zametki, 84:3 (2008), 390–394 | Zbl

[9] Yu.V. Lutsenko, A.N. Skiba, “Konechnye gruppy s subnormalnymi vtorymi ili tretimi maksimalnymi podgruppami”, Matematicheskie zametki, 91:5 (2012), 730–740 | Zbl

[10] Yu.V. Gorbatova, M.N. Konovalova, “Konechnye gruppy s subnormalnymi strogo 2- ili 3-maksimalnymi podgruppami”, Vestnik Omskogo universiteta, 24:3 (2019), 4–12

[11] V.A. Kovaleva, A.N. Skiba, “Finite soluble groups with all $n$-maximal subgroups $\mathfrak{F}$-subnormal”, J. Group Theory, 17:2 (2014), 273–290 | DOI | MR | Zbl

[12] V.S. Monakhov, “O gruppakh s formatsionno subnormalnymi 2-maksimalnymi podgruppami”, Matematicheskie zametki, 105:2 (2019), 269–277 | Zbl

[13] M.N. Konovalova, “Konechnye gruppy s $\mathfrak{F}$-subnormalnymi podgruppami”, Matematicheskie zametki, 108:2 (2020), 215–223 | Zbl

[14] V.S. Monakhov, M.N. Konovalova, “On groups with formational subnormal strictly 2-maximal subgroups”, Ukrains'kyi Matematychnyi Zhurnal, 73:1 (2021), 107–116 | DOI | MR | Zbl

[15] V.S. Monakhov, Vvedenie v teoriyu konechnykh grupp i ikh klassov, Vysheishaya shkola, Minsk, 2006, 207 pp.

[16] L.A. Shemetkov, Formatsii konechnykh grupp, Nauka, M., 1978, 271 pp.

[17] B. Huppert, Endliche Gruppen I, Springer, Berlin–Heidelberg–New York, 1967, 793 pp. | Zbl

[18] T. Connor, D. Leemans, “An atlas of subgroup lattices of finite almost simple groups”, Ars Math. Contemp., 8 (2015), 259–266 | DOI | MR | Zbl

[19] J.H. Conway et al., Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Clarendon Press, Oxford, 1985, 286 pp. | Zbl

[20] T. Dokchitser, GroupNames, (Date of access: 30.04.2021) http://groupnames.org

[21] T.C. Burness, M.W. Liebeck, A. Shalev, “On the length and depth of finite groups”, Proc. London Math. Soc., 119:3 (2019), 1464–1492 | DOI | MR

[22] V.S. Monakhov, “Konechnye gruppy s abnormalnymi i $\mathfrak{U}$-subnormalnymi podgruppami”, Sibirskii matematicheskii zhurnal, 57:2 (2016), 447–462 | Zbl

[23] A.F. Vasilev, S.F. Kamornikov, V.N. Semenchuk, “O reshetkakh podgrupp konechnykh grupp”, Beskonechnye gruppy i primykayuschie algebraicheskie struktury, sb. nauch. st., In-t matem. AN Ukrainy, Kiev, 1993, 27–54

[24] M.N. Konovalova, I.L. Sokhor, “Konechnye gruppy s nekotorymi formatsionno subnormalnymi podgruppami”, Problemy fiziki, matematiki i tekhniki, 2019, no. 4 (41), 51–54

[25] M.N. Konovalova, V.S. Monakhov, “Konechnye gruppy s nekotorymi subnormalnymi 2-maksimalnymi podgruppami”, Problemy fiziki, matematiki i tekhniki, 2020, no. 2 (43), 75–79