Exercise on subgroup of a direct product

Question

Let $G= G_1 \times G_2$ be a direct product of finite groups, let $H$ be a normal subgroup of $G$ with $H \cap G_1= \{1\}$, $H \cap G_2 = \{1\}$.

Prove that $H$ is abelian.

I really don't know how to do this exercise. Hints are also appreciated.

Answer 1

Conjugate $(a,b)\in H$ by $(c,e)$ to get the element $(cac^{-1},b)\in H$.

Can $(a,b)$ and $(cac^{-1},b)$ be distinct elements of $H$?