Two quotient groups implying Cartesian product?

[jostpuur]

Assume that $G$ is some group with two normal subgroups $H_1$ and $H_2$. Assuming that the group is additive, we also assume that $H_1\cap H_2=\{0\}$, $H_1=G/H_2$ and $H_2=G/H_1$ hold. The question is that is $G=H_1\times H_2$ the only possibility (up to an isomorphism) now?

 

[WWGD]

I think if the representation $G= H_1 \times H_2 [/ itex] is unique, then G can be expressed as a semidirect product.$

 

[jostpuur]

I think I managed to prove the claim by using the projections $G\to H_1$ and $G\to H_2$ given by the assumptions, under the assumption $\#G<\infty$, which I did not mention above. Since I was the one asking this, there is probably no need for me to post the full proof, and this will remain as a challenge to the rest. The case $\#G=\infty$ remains open.