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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.

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up vote 0 down vote accepted

We have to use
(Lemma 2, Paragraph 3)

Suppose $G\over G'$ torsion-free then satisfies $D_n$ for any $n>1$. So $G\over G'$ can be factored in a product of three normal subgroups ($A_1\over G'$,...) "of that type". Pick $n=3$. Take $x$ in $G$ and without loss of generality we can suppose $x$ in $A_1$. Then $x$ is in $A_1 A_2$ and also in $A_1 A_3$. So $|A_1 A_2:C_{A_1 A_2}(x)|$ and $|A_1 A_3:C_{A_1 A_3}(x)|$ are finite. But $G=(A_1 A_2) (A_1 A_3)$, so $|G:C_G(x)|$ is finite, so G is FC contrary to the hypothesis.

I think it should work.

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