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Let $G$ be a minimal non-FC-group and suppose $G^*<G$ (where $G^*$ is the finite residual).
Then we have
(i) $G=<G^*, x>$, $x^{p^n} \in G^*$ and $x^p\in Z(G)$,
(ii) $G^*$ is a divisible abelian q-group of finite rank,
(iii) $G^*$ contains no proper infinite subgroup normal in $G$ and $G^*=G'$

Now I have to show, among other things, that if
$HG'=G$,
$H$ is a proper subgroup of $G$,
$G$ satisfies (i), (ii), (iii),
then $H$ is finite.

I work out that $H_G$ is finite ($H_G \cap G^* <G^* \rightarrow |H_G \cap G^*|<\infty \space and \space |{H_G\over {H_G \cap G^*}}|\leq |{G\over G^*}|<\infty$) About $H$?... Any ideas will be appreciated!

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up vote 1 down vote accepted

$H\cap G'$ is normal in $H$ (Isomorphism theorem) and in $G'$ ($G'$ is abelian), so is normal in $HG'$ and the result follow easily from above.

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