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 $G$.