If $G$ is a finite abelian group and $H$ is a subgroup. Show $\exists K\le G$ such that $G/K\cong H$.
I observe that it is enough to find $K\le G$ such that $H\cap K=\{e\}$ and $G=HK$. As in that case $G=H\oplus K$.
But how to find such $K$, can anyone give any idea to do this? Thanks for help in advance.