Let $G$ be a minimal non-FC-group and suppose $G^*
Then we have
(i) $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^*