0
$\begingroup$

Let $G$ be a minimal non-FC-group and suppose $G^* (where $G^*$ is the finite residual).
Then we have
(i) $G=$, $x^{p^n} \in G^*$ 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^* ) About $H$?... Any ideas will be appreciated!

1 Answers 1

1

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