Locally Finite Periodic Groups Saturated with Finite Simple Orthogonal Groups of Odd Dimension

Suppose that $ n $ is an odd integer, $ n\geq 5 $ . We prove that a periodic group  $ G $ , saturated with finite simple orthogonal groups  $ O_{n}(q) $ of odd dimension over fields of odd characteristic, is isomorphic to  $ O_{n}(F) $ for some locally finite field  $ F $ of odd characteristic. In particular, $ G $ is locally finite and countable.