Abstract:
Let ${\goth M}$ be a set of finite groups. Given a group $G$, denote by ${\goth M}(G)$ the set of all subgroups of $G$ isomorphic to the elements of ${\goth M}$. A group $G$ is said to be saturated with groups from ${\goth M}$ (saturated with ${\goth M}$, for brevity) if each finite subgroup of $G$ lies in an element of ${\goth M}(G)$. We prove that a periodic group $G$ saturated with ${\goth M}=\{O_7(q)\mid q\equiv\pm3(\operatorname{mod} 8)\}$ is isomorphic to $O_7(F)$ for some locally finite field $F$ of odd characteristic.
Keywords:periodic group, group of Lie type, orthogonal group, group saturated with a set of groups.