I'm trying to work out this group theory proof:
Suppose that $G$ is a group have that has a normal subgroup $H$ such that $H$ is isomorphic to $D_3$. Prove that exists a subgroup $K$ of $G$ such that $G$ is isomorphic to $H\oplus K$.
Here is what I was thinking:
I was thinking $K$ could be the centralizer of $H$ in $G$. Because then we know that $K$ is a normal subgroup of $G$ since $H$ is normal. Then I was thinking that there would be a relationship between a group $G$ and the direct product of two normal subgroups, however I'm not really sure how to build on that.
Also I was wondering how we can use the fact that $H$ is isomorphic to $D_3$, because I can't really understand how that would help.