The paper deals with locally finite groups $G$ having an involution $\phi$ such that $C_G(\phi)$ is of finite rank. The following theorem gives a very detailed description of such groups.
Let $G$ be a locally finite group having an involution $\phi$ such that $C_G(\phi)$ is of finite rank. Then $G/[G,\phi]$ has finite rank. Furthermore, $[G,\phi]'$ contains a characteristic subgroup $B$ such that the following hold.
(1) $B$ is a product of finitely many subgroups normal in $[G,\phi]$ isomorphic to either PSL$(2,K)$ or SL$(2,K)$ for some infinite locally finite fields $K$ of odd characteristic.
(2) $[G,\phi]'/B$ has finite rank.