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.