Periodic Groups with One Finite Nontrivial Sylow 2-Subgroup

D. V. Lytkina ; V. D. Mazurov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 4, pp. 146-154

Résumé

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 the direct product of $m$ dihedral groups, where $m\le d$. Then $G$ is finite (and isomorphic to the 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 the 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 the 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 $2^{2d}$.

Détail

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

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