Let $G$ and $H$ are two divisible groups that each of which is is isomorphic to a subgroup of the other, then $G\cong H$.
What I've done is to use the injective property for both groups:
$G\cong K\le H$ so we have $G\stackrel{\iota}{\hookrightarrow} H$ and $G\stackrel{id}{\longrightarrow} G$ and then there exists $H \stackrel{\phi}{\longrightarrow} G$ that $\phi\circ i=id|_G$.
$H\cong S\le G$ so we have $H\stackrel{\iota}{\hookrightarrow} G$ and $H\stackrel{id}{\longrightarrow} H$ and then there exists $G \stackrel{\psi}{\longrightarrow} H$ that $\psi\circ i=id|_H$.
Is my approach right? May I ask you what will be happen if we omit the adjective divisible? Thanks.