Let $G$ be a minimal non-FC-group with $G'<G$.
Suppose that $G$ has no non-trivial finite factor group (absurdum hypothesis).
Now, $G\over G'$ is a divisible abelian group; but also periodic?
I need periodicity to make uses of the following lemma:
Let G be a minimal non-FC-group with no non-trivial finite factor groups. Let $G\over K$ be a periodic factor group of $G$ with a homomorphic image $G\over H$ isomorphic to a Prüfer group. Then $G\over K$ is a p-group.