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 or its dual for some ), 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.
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The Structure of Locally Compact Abelian Groups,
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P.-E. Caprace and N. Monod,
Decomposing locally compact groups into simple pieces,
Math. Proc. Cambridge Philos. Soc. 150 (2011), no. 1, 97–128.