Geodesic
Finite Groups with Three Nonconjugate Maximal Formational Subgroups
Matematičeskie zametki, Tome 111 (2022) no. 3, pp. 354-364.

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

A constructive description is obtained for the hereditary $Z$-saturated formations $\mathfrak{F}$ of finite solvable groups containing every solvable group possessing three pairwise nonconjugate maximal subgroups belonging to $\mathfrak{F}$. It is proved that a finite group $G$ is supersolvable if it has three pairwise nonconjugate supersolvable maximal subgroups and its commutator subgroup $G'$ is nilpotent.
Keywords: finite group, maximal subgroup, $Z$-saturated formation, formation with the Belonogov property, formation with the Kegel property
Mots-clés : supersolvable group. 
@article{MZM_2022_111_3_a2,
     author = {A. F. Vasil'ev and V. I. Murashka and A. K. Furs},
     title = {Finite {Groups} with {Three} {Nonconjugate} {Maximal} {Formational} {Subgroups}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {354--364},
     publisher = {mathdoc},
     volume = {111},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_111_3_a2/}
}
TY  - JOUR
AU  - A. F. Vasil'ev
AU  - V. I. Murashka
AU  - A. K. Furs
TI  - Finite Groups with Three Nonconjugate Maximal Formational Subgroups
JO  - Matematičeskie zametki
PY  - 2022
SP  - 354
EP  - 364
VL  - 111
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2022_111_3_a2/
LA  - ru
ID  - MZM_2022_111_3_a2
ER  - 
%0 Journal Article
%A A. F. Vasil'ev
%A V. I. Murashka
%A A. K. Furs
%T Finite Groups with Three Nonconjugate Maximal Formational Subgroups
%J Matematičeskie zametki
%D 2022
%P 354-364
%V 111
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2022_111_3_a2/
%G ru
%F MZM_2022_111_3_a2
A. F. Vasil'ev; V. I. Murashka; A. K. Furs. Finite Groups with Three Nonconjugate Maximal Formational Subgroups. Matematičeskie zametki, Tome 111 (2022) no. 3, pp. 354-364. http://geodesic.mathdoc.fr/item/MZM_2022_111_3_a2/

[1] V. A. Belonogov, “Konechnye gruppy s edinstvennym klassom nenilpotentnykh maksimalnykh podgrupp”, Sib. matem. zhurn., 5:5 (1964), 987–995 | MR | Zbl

[2] V. A. Belonogov, “Konechnye gruppy s paroi nesopryazhennykh nilpotentnykh maksimalnykh podgrupp”, Dokl. AN SSSR, 161:6 (1965), 1255–1256 | MR | Zbl

[3] V. N. Semenchuk, “Konechnye gruppy s sistemami minimalnykh ne $\mathfrak{F}$-grupp”, Podgruppovoe stroenie konechnykh grupp, Nauka i tekhnika, Minsk, 1981, 138–148

[4] A. F. Vasilev, “K probleme perechisleniya lokalnykh formatsii s zadannym svoistvom”, Voprosy algebry, 1987, no. 3, 3–11

[5] A. F. Vasilev, “O perechislenii lokalnykh formatsii s usloviem Kegelya”, Voprosy algebry, 1993, no. 7, 86–93

[6] A. Ballester-Bolinches, L. M. Ezquerro, “On formations with the Kegel property”, J. Group Theory, 5 (2005), 605–611 | DOI | MR | Zbl

[7] O. H. Kegel, “Zur Struktur mehrfach factorisierbarer endlicher Gruppen”, Math. Z., 87:1 (1965), 42–48 | DOI | MR | Zbl

[8] L. S. Kazarin, “On groups which are the product of two solvable subgroups”, Comm. Algebra, 14:6 (1986), 1001–1066 | DOI | MR | Zbl

[9] L. S. Kazarin, “Faktorizatsii konechnykh grupp razreshimymi podgruppami”, Ukr. matem. zhurn., 43:7-8 (1991), 947–950 | MR

[10] L. A. Shemetkov, A. N. Skiba, Formatsii algebraicheskikh sistem, Sovremennaya algebra, Nauka, M., 1989 | MR | Zbl

[11] L. A. Shemetkov, “Frattini extensions of finite groups and formations”, Comm. Algebra, 23:3 (1997), 955–964 | DOI | MR

[12] A. Ballester-Bolinches, M. D. Perez-Ramos, “On a question of L. A. Shemetkov”, Comm. Algebra, 27:11 (1999), 5615–5618 | DOI | MR | Zbl

[13] K. Doerk, T. Hawkes, Finite Soluble Groups, De Gruyter Exp. Math., 4, Walter de Gruyter, Berlin, 1992 | MR

[14] A. F. Vasilev, V. I. Murashko, “Formatsii i proizvedeniya $\mathrm F(G)$-subnormalnykh podgrupp konechnykh razreshimykh grupp”, Matem. zametki, 107:3 (2020), 376–390 | DOI | MR

[15] A. N. Skiba, “Ob odnom klasse lokalnykh formatsii konechnykh grupp”, Dokl. AN BSSR, 34:11 (1990), 982–984 | MR

[16] V. N. Semenchuk, A. F. Vasilev, “Kharakterizatsiya lokalnykh formatsii F po zadannym svoistvam minimalnykh ne $\mathfrak{F}$-grupp”, Issledovanie normalnogo i podgruppovogo stroeniya konechnykh grupp. Trudy Gomelskogo seminara, Nauka i tekhnika, Minsk, 1984, 175–181 | MR

[17] A. Ballester-Bolinches, L. M. Ezquerro, Classes of Finite Groups, Math. Appl. (Springer), 584, Springer, Dordrecht, 2006 | MR | Zbl

[18] B. Huppert, Endliche Gruppen. I, Grundlehren Math. Wiss., 134, Springer-Verlag, Berlin, 1967 | MR | Zbl

[19] R. L. Griess, P. Schmid, “The Frattini module”, Arch. Math. (Basel), 30:3 (1978), 256–266 | DOI | MR | Zbl

[20] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson,, $\mathbb{ATLAS}$ of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups, Clarendon Press, Oxford, 1985 | MR

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

[22] A. Ballester-Bolinches, M. D. Perez-Ramos, “On $\mathfrak{F}$-critical groups”, J. Algebra, 147:3 (1995), 948–958 | DOI | MR

[23] L. S. Kazarin, A. Martínez-Pastor, M. D. Pérez-Ramos, “Finite trifactorsed groups and $\pi$-decomposability”, Bull. Aust. Math. Soc., 97:2 (2018), 218–228 | DOI | MR | Zbl