Monolith not is divisibility-closed in abelian group

Statement
It is possible to have an abelian group $$G$$ that is a monolithic group (i.e., it has a unique minimal normal subgroup that is contained in every nontrivial normal subgroup) but such that the monolith (the unique minimal normal subgroup, which is therefore also the socle) is not a divisibility-closed subgroup of $$G$$.

In particular, this also shows that the socle in an abelian group need not be a divisibility-closed subgroup.

Related facts

 * Fully invariant subgroup of abelian group not implies divisibility-closed
 * Finite subgroup of abelian group not implies divisibility-closed

Proof
Let $$p$$ be a prime number and let $$G$$ be the $$p$$-quasicyclic group. $$G$$ is $$p$$-divisible. Its monolith is the unique subgroup of order $$p$$, which is not $$p$$-divisible. Thus, the monolith is not divisibility-closed in $$G$$.