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
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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.
