  We classify the locally compact second-countable (l.c.s.c.) groups $A$ that
are abelian and topologically characteristically simple. All such groups $A$
occur as the monolith of some soluble l.c.s.c. group $G$ of derived length at
most $3$; with known exceptions (specifically, when $A$ is $\mathbb{Q}^n$ or
its dual for some $n \in \mathbb{N}$), we can take $G$ to be compactly
generated. This amounts to a classification of the possible isomorphism types
of abelian chief factors of l.c.s.c. groups, which is of particular interest
for the theory of compactly generated locally compact groups.
