[nooonuii]

Since $H$ is a subgroup of $<a>$, $H=<a^k>$ for some $k$. The following identity may help
$$ga^ng^{-1}=(gag^{-1})^n.$$

Interesting Problem : Is the following statement true?

If $K$ is a normal subgroup of $H$ and $H$ is a normal subgroup of $G$ then $K$ is a normal subgroup of $G$.