Groups saturated with finite simple groups $L_3(2^n)$ and $L_4(2^l)$

A. A. Shlepkin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 2, pp. 249-257

Résumé

Let $\mathfrak{M}$ be a certain set of groups. For a group $G$, we denote by $\mathfrak{M}(G)$ the set of all subgroups of $G$ that are isomorphic to elements of $\mathfrak{M}$. A group $G$ is said to be saturated with groups from $\mathfrak{M}$ if any finite subgroup of $G$ is contained in some element of $\mathfrak{M}(G)$. We prove that if $G$ is a periodic group or a Shunkov group and $G$ is saturated with groups from the set $\{L_3(2^n), L_4(2^l)\mid n=1,2,\ldots; l=1,\ldots, l_0\},$ where $l_0$ is fixed, then the set of elements of finite order from $G$ forms a group isomorphic to one of the groups from the set $\{L_3 (R), L_4(2^l)\mid l=1,\ldots, l\}$, where $R$ is an appropriate locally finite field of characteristic $2$.

Détail

Keywords: periodic group, Shunkov group, saturation of a group with a set of groups.

Bibliographie

[1] Belonogov V.A., Zadachnik po teorii grupp, Nauka, M., 2000, 464 pp.

[2] Ditsman A.P., “O $p$-gruppakh”, Dokl. AN SSSR, 1937, no. 15, 71–76

[3] Kargapolov M.I., Merzlyakov Yu.I., Osnovy teorii grupp, Lan, SPb., 2009, 287 pp. | MR

[4] Kondratev A.S., Mazurov V.D., “2-signalizatory konechnykh prostykh grupp”, Algebra i logika, 42:5 (2003), 594–623 | MR

[5] Unsolved problems in group theory, The Kourovka notebook, no. 19, eds. V. D. Mazurov and E. I. Khukhro, Inst. Math. SO RAN Publ., Novosibirsk, 2018, 250 pp. URL: https://kourovka-notebook.org/ | MR

[6] Lytkina D.V., Mazurov V.D., “Periodicheskie gruppy, nasyschennye gruppami $L_3(2^m)$”, Algebra i logika, 46:5 (2007), 606–626 | MR | Zbl

[7] Maslova N.V., Belousov I.N., Minigulov N.A., “Otkrytye problemy, sformulirovannye na XII shkole-konferentsii po teorii grupp, posvyaschennoi 85-letiyu V.A. Belonogova”, Tr. In-ta matematiki i mekhaniki UrO RAN, 26:3 (2020), 275–285 | DOI | MR

[8] Sanov I.N., “Reshenie problemy Bernsaida dlya perioda 4”, Uchen. zapiski LGU. Ser. Matematicheskaya, 1940, no. 55, 166–170 | MR

[9] Senashov V.I., Shunkov V.P., Gruppy s usloviyami konechnosti, Izd-vo SO RAN, Novosibirsk, 2001, 326 pp. | MR

[10] Senashov V.I., “On periodic groups of Shunkov with the Chernikov centralizers of involutions”, Izv. Irkut. gos. un-ta. Ser. Matematika, 32 (2020), 101–117 | DOI | MR | Zbl

[11] Senashov V.I., “On periodic Shunkovs groups with almost layer-finite normalizers of finite subgroups”, Izv. Irkut. gos. un-ta. Ser. Matematika, 37 (2021), 118–132 | DOI | MR | Zbl

[12] Suprunenko D.A., Gruppy matrits, Nauka, M., 1972, 352 pp. | MR

[13] Cherep A.A., “O elementakh konechnogo poryadka v biprimitivno konechnykh gruppakh”, Algebra i logika, 26:4 (1987), 518–521 | MR

[14] Shlepkin A.A., “Gruppy Shunkova, nasyschennye lineinymi i unitarnymi gruppami stepeni 3 nad polyami nechetnykh poryadkov”, Sib. elektron. mat. izv., 13 (2016), 341–351 | DOI | MR | Zbl

[15] Shlepkin A.A., Sabodakh I.V., “O dvukh svoistvakh gruppy Shunkova”, Izv. Irkut. gos. un-ta. Ser. Matematika, 35 (2021), 103–119 | DOI | MR | Zbl

[16] Shlepkin A.K., “O sopryazhenno biprimitivno konechnykh gruppakh s usloviem primarnoi minimalnosti”, Algebra i logika, 22:2 (1983), 232–231 | MR

[17] Shlepkin A.K., “O nekotorykh periodicheskikh gruppakh, nasyschennykh konechnymi prostymi gruppami”, Mat. tr., 1:1 (1998), 129–138 | MR | Zbl

[18] Shunkov V.P., “O periodicheskikh gruppakh s pochti regulyarnoi involyutsiei”, Algebra i logika, 11:4 (1972), 470–494