$A\cap G=A\cap H=1$ implies $A=1$? Suppose $G\times H$ is an abelian group and $A\le G\times H$.
Is it true that $A\cap G=A\cap H=1$ implies $A=1$? ($1$ is the trivial subgroup)

In general, no. Take for example $\langle (1,1) \rangle \leq \mathbb{Z}_2 \times \mathbb{Z}_2$.
