I am trying to prove this:

If a divisible group $E$ containining $A$ is minimal divisible then $A$ is an essential subgroup of $E$.

Let $ < c > =C, \ C\cap A = 0$. Without loss of generality we can say that $ o (c) = \infty$ or $o (c) = p$.

Now I want to say that $C$ can be embedded in a subgroup $B \subset E$, such that $B \simeq \mathbb{Q} $, or $B \simeq \mathbb{Z}_{p^\infty }$. But I can't figure out how to proove it.