A.A. Shlepkin. Groups saturated with finite simple groups $L_3(2^n)$ and $L_4(2^l)$ ... P. 249-257

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 of the set $\{L_3 (R), L_4(2^l)\mid l=1,\ldots, l\}$, where $R$ is an appropriate locally finite field of characteristic $2$. %The work is dedicated to the fond memory of Vyacheslav Aleksandrovich Belonogov.

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

Received January 8, 2022

Revised March 20, 2022

Accepted March 28, 2022

Funding Agency: This work was supported by the Russian Science Foundation (project no. 19-71-10017).

Aleksei Anatolievich Shlepkin, Cand. Sci. (Phys.-Math.), Siberian Federal University, Krasnoyarsk, 660041 Russia, e-mail: shlyopkin@mail.ru

REFERENCES

1.   Belonogov V.A. Zadachnik po teorii grupp [Exercise book on group theory]. Moscow: Nauka Publ., 2000, 464 p.

2.   Dietzmann A.P. On p-groups. Dokl. Acad. Nauk SSSR, 1937, vol. 15, pp. 71–76 (in Russian).

3.   Kargapolov M.I., Merzljakov Yu.I. Fundamentals of the theory of groups. NY; Heidelberg; Berlin: Springer-Verlag, 1979, 203 p. ISBN: 978-1-4612-9966-0 . Original Russian text published in Kargapolov M.I., Merzlyakov Yu.I. Osnovy teorii grupp, St. Petersburg: Lan’ Publ., 2009, 287 p.

4.   Kondrat’ev A.S., Mazurov V.D. 2-signalizers of finite simple groups. Algebra and Logic, 2003, vol. 42, no. 5, pp. 333–348. doi: 10.1023/A:1025923522954 

5.   The Kourovka notebook: Unsolved problems in group theory. No. 19, ed. by V. D. Mazurov and E. I. Khukhro, Novosibirsk: Inst. Math. SO RAN Publ., 2018, 250 p. Available on: https://kourovka-notebook.org/ 

6.   Lytkina D.V., Mazurov V.D. Periodic groups saturated with $L_3(2^m)$. Algebra and Logic, 2007, vol. 46, no. 5, pp. 330–340. doi: 10.1007/s10469-007-0033-z 

7.   Maslova N.V., Belousov I.N., Minigulov N.A. Open questions formulated at the 13th school-conference on group theory dedicated to V.A. Belonogov’s 85th birthday. Trudy Inst. Mat. i Mekh. UrO RAN, 2020, vol. 26, no. 3, pp. 275–285 (in Russian).

8.   Sanov I.N. Solution of the Burnside problem for exponent 4. Uchen. Zap. Leningr. Univ. Ser. Mat., 1940, no. 10, pp. 166–170 (in Russian).

9.   Senashov V.I., Shunkov V.P. Gruppy s usloviyami konechnosti [Groups with finiteness conditions]. Novosibirsk: SO RAN Publ., 2001, 326 p. ISBN: 5-7692-0439-7 .

10.   Senashov V.I. On periodic groups of Shunkov with the Chernikov centralizers of involutions. The Bulletin of Irkutsk State University. Ser. Mathematics, 2020, vol. 32, pp. 101–117. doi: 10.26516/1997-7670.2020.32.101 

11.   Senashov V.I. On periodic Shunkov’s groups with almost layer-finite normalizers of finite subgroups. The Bulletin of Irkutsk State University. Ser. Mathematics, 2021, no. 37, pp. 118–132. doi: 10.26516/1997-7670.2021.37.118 

12.   Suprunenko D.A. Matrix groups. Providence: AMS, 1976, 252 p. ISBN: 0821813412 . Original Russian text published in Suprunenko D.A. Gruppy matrits, Moscow: Nauka Publ., 1972, 352 p.

13.   Cherep A.A. Set of elements of finite order in a biprimitively finite group. Algebra and Logic, 1987, vol. 26, no. 4, pp. 311–313. doi: 10.1007/BF01980245 

14.   Shlepkin A.A. On Shunkov groups, saturated with linear and unitary groups of dimension 3 over fields of odd orders. Sib. Elektron. Mat. Izv., 2016, vol. 13, pp. 341–351 (in Russian). doi: 10.17377/semi.2016.13.029 

15.   Shlepkin A.A., Sabodakh I.V. On two properties of Shunkov group. The bulletin of Irkutsk State University. Series Mathematics, 2021, no. 35, pp. 103–119. doi: 10.26516/1997-7670.2021.35.103 

16.   Shlepkin A.K. Conjugately biprimitively finite groups with the primary minimal condition. Algebra Logic, 1983, vol. 22, no. 2, pp. 165–169. doi: 10.1007/BF01978669 

17.   Shlepkin A. On certain torsion groups saturated with finite simple groups. Sib. Adv. Math., 1999, vol. 9, no. 2, pp. 100–108.

18.   Shunkov V.P. On periodic groups with an almost regular involution. Algebra and Logic, 1972, vol. 11, no. 4, pp. 260–272. doi: 10.1007/BF02219098 

Cite this article as: A.A. Shlepkin. Groups saturated with finite simple groups $L_3(2^n)$ and $L_4(2^l)$. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 2, pp. 249–257.