Periodic groups saturated with the groups $L_2(p^n)$

Given an indexing set $I$ and a finite field $K_\alpha$ for each $\alpha\in I$, $\mathfrak R=\{L_2(K_\alpha)|\alpha\in I\}$ and $\mathfrak N=\{SL_2(K_\alpha)|\alpha\in I\}$. We prove that each periodic group $G$ saturated with groups in $\mathfrak R(\mathfrak N)$ is isomorphic to $L_2(P)$ (respectively $SL_2(P)$) for a suitable locally finite field $P$.