Normalizer/Centralizer theorem

This is exercise 3.2.24 from Scott, Group Theory.

If $H$ is a finite maximal abelian normal subgroup of $G$ and $K$ is a normal abelian subgroup of $G$, then $K$ is finite.

The hint is to use Normalizer/Centralizer theorem.

Solutions Collecting From Web of "Normalizer/Centralizer theorem"

Since $H$ is finite, $Aut(H)$ is finite. By the Normalizer/Centralizer theorem, $\frac{N_{G}(H)}{C_{G}(H) }= \frac{G}{C_{G}(H)} \ $ is isomorphic to a subgroup of $Aut(H)$ and so is finite. Now we note that if $M \ \trianglelefteq \ G \ $, $M$ is abelian and $M \leq C_{G}(H) \ $, then $HM$ is abelian and normal in $G$, but $H$ is maximal abelian normal so $HM\leq H \ $ and then $M\leq H \ $. Note that
$K \cap C_{G}(H) \ $ is abelian because $K$ is abelian and $K \cap C_{G}(H) \trianglelefteq G \ $ because $K \trianglelefteq G \ $ and $C_{G}(H) \trianglelefteq G$. Then $K \cap C_{G}(H) \subseteq H \ $ and so $K \cap C_{G}(H) \ $ is finite because $H$ is finite. $\frac{KC_{G}(H)}{C_{G}(H)} \simeq \frac{K}{K \cap C_{G}(H)} \ $ is a subgroup of $\frac{G}{C_{G}(H)} \ $ and so is finite. Then $|K| = |\frac{K}{K \cap C_{G}(H)}| \cdot |K \cap C_{G}(H)| \ $ is finite.