On critical $\sigma$-local formations of finite groups

I. N. Safonova

Trudy Instituta matematiki, Tome 31 (2023) no. 2, pp. 63-80.

Résumé

In this article we study minimal $\sigma$-local non-$\mathfrak H$-formations of finite groups (or, in other words, $\mathfrak H_\sigma$-critical formations), i. e. such $\sigma$-local formations not included in the class of groups $\mathfrak H$, all of whose proper $\sigma$-local subformations are contained in $\mathfrak H$. A description of minimal $\sigma$-local non$\mathfrak H$-formations for an arbitrary $\sigma$-local formation $\mathfrak H$ of classical type is obtained (а $\sigma$-local formation is called a $\sigma$-local formation of classical type if it has a $\sigma$-local definition such that all its non-Abelian values are $\sigma$-local). The main result of the work in the class of $\sigma$-local formations solves the problem of L. A. Shemetkov (1980) on the description of critical formations for given classes of finite groups. As corollaries, descriptions of $\mathfrak H_\sigma$-critical formations are given for a number of specific classes of finite groups, such as the classes of all $\sigma$-nilpotent, meta-$\sigma$-nilpotent groups, as well as the class all groups with $\sigma$-nilpotent commutator subgroup.

Bibliographie

[1] Shemetkov L. A., “Ekrany stupenchatykh formatsii”, Tr. VI Vsesoyuzn. simpoziuma po teorii grupp, Naukova dumka, Kiev, 1980, 37–50

[2] Skiba A. N., “O kriticheskikh formatsiyakh”, Izv. AN BSSR. Ser. fiz.-mat. nauk, 1980, no. 4, 27–33 | Zbl

[3] Skiba A. N., “O kriticheskikh formatsiyakh”, Beskonechnye gruppy i primykayuschie algebraicheskie struktury, In-t matematiki AN Ukrainy, Kiev, 1993, 258–268

[4] Shemetkov L. A., Skiba A. N., Formatsii algebraicheskikh sistem, Nauka, M., 1989 | MR

[5] Skiba A. N., Algebra formatsii, Belaruskaya navuka, Minsk, 1997 | MR

[6] Safonov V. G., “O minimalnykh kratno lokalnykh ne $\mathfrak{H}$-formatsiyakh konechnykh grupp”, Voprosy algebry, 1995, no. 8, 109–138 | Zbl

[7] Safonov V. G., “O kriticheskikh kratno lokalnykh formatsiyakh konechnykh grupp”, Ves. Akad. navuk Belarusi. Ser. fiz.-mat. navuk, 1997, no. 3, 57–61

[8] Dzharadin D., “Minimalnye $p$-nasyschennye nenilpotentnye formatsii”, Voprosy algebry, 1995, no. 8, 59–64

[9] Safonova I. N., “O minimalnykh $\omega$-lokalnykh nesverkhrazreshimykh formatsiyakh”, Voprosy algebry, 1998, no. 12, 123–130 | MR | Zbl

[10] Safonova I. N., “O minimalnykh $\omega$-lokalnykh ne $\mathfrak H$-formatsiyakh”, Ves. Nats. akad. navuk Belarusi. Ser. fiz.-mat. navuk, 1999, no. 2, 23–27 | MR

[11] Safonova I. N., “K teorii $\mathfrak{H}_{l^{\omega}}$-kriticheskikh formatsii konechnykh grupp”, Izv. Gomelskogo gos. un-ta, 2001, no. 3(6), 124–133

[12] Selkin V. M., “Minimalnye nasledstvennye $\omega$-lokalnye ne $\mathfrak H$-formatsii”, Ukrainskii matematicheskii zhurnal, 54:3 (2002), 373–380 | MR | Zbl

[13] Safonova I. N., “O kriticheskikh chastichno nasyschennykh formatsiyakh konechnykh grupp”, Ves. Nats. akad. navuk Belarusi. Ser. fiz.-mat. navuk, 2006, no. 2, 51–55

[14] Selkin V. M., “O minimalnykh $\tau$-zamknutykh $\omega$-lokalnykh ne $\mathfrak H$-formatsiyakh”, Tr. In-ta matematiki, 16:1 (2008), 81–85 | Zbl

[15] Ryabchenko A. I., “K teorii chastichno nasyschennykh formatsii”, Izv. Gomelskogo gos. un-ta, 2008, no. 6(51)-2, 153–160

[16] Safonov V. G., Safonova I. N., “O minimalnykh totalno $\omega$-nasyschennykh nenilpotentnykh formatsiyakh konechnykh grupp”, Vestn. Vitebskogo gos. un-ta, 2014, no. 6(84), 9–15

[17] Vedernikov V. A., Sorokina M. M., “Kompozitsionnye nasledstvennye kriticheskie formatsii”, Voprosy algebry, 1997, no. 11, 59–64

[18] Sorokina M. M., “O kompozitsionnykh normalno nasledstvennykh kriticheskikh formatsiyakh”, Voprosy algebry, 1998, no. 12, 23–36

[19] Bliznets I. V., Skiba A. N., “O $\mathfrak{H}_{\Theta^{\mathfrak{L}}}$-kriticheskikh formatsiyakh”, Izv. Gomelskogo gos. un-ta, 1999, no. 1, 140–144 | Zbl

[20] Bliznets I. V., “Kriticheskie $\omega$-kompozitsionnye formatsii”, Ves. Nats. akad. navuk Belarusi. Ser. fiz.-mat. navuk, 2002, no. 4, 115–117 | Zbl

[21] Zadorozhnyuk M. V., “O minimalnykh $\tau$-zamknutykh $\omega$-kompozitsionnykh nenilpotentnykh formatsiyakh”, Ves. Nats. akad. navuk Belarusi. Ser. fiz.-mat. navuk, 2006, no. 3, 21–24 | MR

[22] Belous L. I., Selkin V. M., Skiba A. N., “Ob odnom klasse kriticheskikh $\omega$-kompozitsionnykh formatsii”, Dokl. Nats. akad. nauk Belarusi, 50:6 (2006), 36–40 | MR | Zbl

[23] Belous L. I., Sel'kin V. M., “On minimal $\omega$-composition non-$\mathfrak{H}$-formations”, Algebra and discrete mathematics, 2006, no. 4, 1–11 | MR | Zbl

[24] Zhiznevskii P. A., Safonov V. G., “O kriticheskikh chastichno kompozitsionnykh formatsiyakh”, Ves. Nats. akad. nauk Belarusi. Ser. fiz.-mat. navuk, 2010, no. 3, 44–49 | MR

[25] Zhiznevskii P. A., “O suschestvovanii kriticheskikh formatsii”, Dokl. Nats. akad. nauk Belarusi, 55:1 (2011), 27–30 | MR | Zbl

[26] Safonov V. G., “$\mathfrak{H}^{\tau}_{\infty}$-kriticheskie formatsii”, Izv. Gomelskogo gos. un-ta, 2008, no. 2(47), 169–176

[27] Safonov V. G., “O suschestvovanii minimalnykh $\tau$-zamknutykh totalno nasyschennykh ne $\mathfrak{H}$-formatsii”, Tr. In-ta matematiki, 16:1 (2008), 67–72 | MR | Zbl

[28] Vedernikov V. A., Koptyukh D. G., “Nasledstvennye kriticheskie $\Omega$-kompozitsionnye formatsii”, Ukr. mat. zhurn., 53:5 (2001), 579–588 | MR | Zbl

[29] Sorokina M. M., Silenok N. V., “Kriticheskie $\Omega$-rassloennye formatsii konechnykh grupp”, Matematicheskie zametki, 72:2 (2002), 269–282 | DOI | MR | Zbl

[30] Korpacheva M. A., Sorokina M. M., “O kriticheskikh $\omega$-veernykh formatsiyakh konechnykh grupp”, Matematicheskie zametki, 79:1 (2006), 87–94 | DOI | MR | Zbl

[31] Safonov V. G., Safonova I. N., “Privodimye $\omega$-nasyschennye formatsii s razreshimym defektom $\leqslant 2$”, Izv. Gomelskogo gos. un-ta, 2005, no. 5(32), 162–165

[32] Safonov V. G., Safonova I. N., “O kommutativnykh polugruppakh razreshimykh totalno $\omega$-nasyschennykh formatsii”, Problemy fiziki, matematiki i tekhniki, 2015, no. 4(25), 80–86 | Zbl

[33] Safonova I. N., “O minimalnykh $\sigma$-lokalnykh ne $\mathfrak H$-formatsiyakh konechnykh grupp”, Problemy fiziki, matematiki i tekhniki, 2020, no. 4(45), 105–112

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

[35] Skiba A. N., “On one generalization of the local formations”, Probl. Phys. Math. Tech., 2018, no. 1(34), 79–82 | MR | Zbl

[36] Chi Z., Safonov V. G., Skiba A. N., “On one application of the theory of $n$-multiply $\sigma$-local formations of finite groups”, Probl. Phys. Math. Tech., 2018, no. 2(35), 85–88 | MR | Zbl

[37] Chi Z., Safonov V. G., Skiba A. N., “On $n$-multiply $\sigma$-local formations of finite groups”, Comm. Algebra, 47:3 (2019), 957–968 | DOI | MR | Zbl

[38] Tsarev A., “Laws of the lattices of $\sigma$-local formations of finite groups”, Mediterranean Journal of Mathematics, 17:3 (2020), 75 | DOI | MR | Zbl

[39] Safonova I. N., Safonov V. G., “On some properties of the lattice of totally $\sigma$-local formations of finite groups”, Journal of the Belarusian State University. Mathematics and Informatics, 2020, no. 3, 6–16 | DOI | MR

[40] Vorobev N. N., Staselko I. I., Khodzhagulyev A. O., “Otdelimye reshetki kratno $\sigma$-lokalnykh formatsii”, Sib. mat. zhurn., 62:4 (2021), 721–735

[41] Safonova I. N., “A criterion for $\sigma$-locality of a non-empty formation”, Comm. Algebra, 50:6 (2022), 2366–2376 | DOI | MR | Zbl

[42] Safonova I. N., “On properties of the lattice of all $\tau$-closed $n$-multiply $\sigma$-local formations”, Comm. Algebra, 51:10 (2023), 4454–4461 | DOI | MR | Zbl

[43] Safonova I. N., “On $\sigma$-inductive lattices of $n$-multiply $\sigma$-local formations of finite groups”, J. Algebra and its Applications | DOI | MR