D.V. Lytkina, V.D. Mazurov. On periodic groups with a finite nontrivial Sylow 2-subgroup ... P. 146-154

The following results are proved. Let $d$ be a natural number, and let $G$ be a group of finite even exponent such that each of its finite subgroups is contained in a subgroup isomorphic to a direct product of $m$ dihedral groups, where $m\leqslant d$. Then $G$ is finite (and isomorphic to a direct product of at most $d$ dihedral groups). Next, suppose that $G$ is a periodic group and $p$ is an odd prime. If every finite subgroup of $G$ is contained in a subgroup isomorphic to a direct product $D_1\times D_2$, where $D_i$ is a dihedral group of order $2p^{r_i}$ with natural $r_i$, $i=1,2$, then $G=M_1\times M_2$, where $M_i=\langle H_i,t\rangle$, $t_i$ is an element of order $2$, $H_i$ is a locally cyclic $p$-group, and $h^{t_i}=h^{-1}$ for every $h\in H_i$, $i=1,2$. Now, suppose that $d$ is a natural number and $G$ is a solvable periodic group such that every of its finite subgroups is contained in a subgroup isomorphic to a direct product of at most $d$ dihedral groups. Then $G$ is locally finite and is an extension of an abelian normal subgroup by an elementary abelian $2$-subgroup of order at most $2^{2d}$.

Keywords: periodic group, exponent, Sylow 2-subgroup, dihedral group, direct product, saturating set

Received May 5, 2023

Revised June 21, 2023

Accepted June 26, 2023

Funding Agency: This work was supported by the Program for Fundamental Research of the Russian Academy of Sciences (project no. FWNF-2022-0002).

Daria Viktorovna Lytkina, Dr. Phys.-Math. Sci., Prof., Siberian State University of Telecommunications and Information Sciences, Novosibirsk, 630102 Russia, e-mail: daria.lytkin@gmail.com

Victor Danilovich Mazurov, Dr. Phys.-Math. Sci., Corresponding Member RAS, Sobolev Institute of Mathematics of the Siberian Branch of the RAS, Novosibirsk, 630090; Siberian State University of Telecommunications and Information Sciences, Novosibirsk, 630102 Russia, e-mail: vic.mazurov@gmail.com

REFERENCES

1.   Ivanov S. V. The free Burnside groups of sufficiently large exponents. Internat. J. Algebra Comput., 1994, vol. 4, no. 1–2, pp. 1–308. doi: 10.1142/S0218196794000026

2.   Lysenok I. G. Infinite Burnside groups of even exponent. Izv. Math., 1996, vol. 60, no. 3, pp. 453–654. doi: 10.1070/IM1996v060n03ABEH000077

3.   Ivanov S. V., Olshanskii A. Yu. On finite and locally finite subgroups of free Burnside groups of large even exponents. J. Algebra, 1997, vol. 195, no. 1, pp. 241–284. doi: 10.1006/jabr.1996.6941

4.   Shlepkin A. K., Rubashkin A. G. A class of periodic groups. Algebra and Logic, 2005, vol. 44, no. 1, pp. 65–71. doi: 10.1007/s10469-005-0008-x

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

6.   Amberg B., Kazarin L. Periodic groups saturated by dihedral subgroups. In: Proc. Conf. Ischia group theory 2010, World Sci. Publ., 2012, pp. 11–19. doi: 10.1142/9789814350051_0002

7.   Belousov I. N., Kondrat’ev A. S., Rozhkov A. V. The 12th school–conference on group theory dedicated to the 65th birthday of A.A. Makhnev (informational paper). Trudy Inst. Mat. Mekh. UrO RAN, 2018, vol. 24, no. 3, pp. 286–295 (in Russian). doi: 10.21538/0134-4889-2018-24-3-286-295

8.   Kukharev A. V., Shlepkin A. A. Locally finite groups saturated with direct product of two finite dihedral groups. Izvestiya Irkutsk. Gos. Univ., Ser. Matem., 2023, vol. 44, pp. 71–81 (in Russian). doi: 10.26516/1997-7670.2023.44.71

9.   Shunkov V. P. On periodic groups with almost regular involution. Algebra i Logika, 1972, vol. 11, no. 4, pp. 470–493 (in Russian).

10.   Lytkina D. V., Tukhvatullina L. R., Filippov K. A. The periodic groups saturated by finitely many finite simple groups. Siberian Math. J., 2008, vol. 49, no. 2, pp. 317–321. doi: 10.1007/s11202-008-0031-y

11.   Lytkina D. V., Mazurov V. D. Fusion of 2-elements in periodic groups with finite Sylow 2-subgroups. Sib. Electron. Math. Reports, 2020, vol. 17, pp. 1953–1958. doi: 10.33048/semi.2020.17.131

12.   Hall M. The theory of groups, NY, Macmillan, 1959, 434 p. Translated to Russian under the title Teoriya grupp, Moscow, Inostr. Liter. Publ., 1962, 467 p.

Cite this article as: D.V. Lytkina, V.D. Mazurov. On periodic groups with a finite nontrivial Sylow 2-subgroup. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 146–154; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S160–S167.