Two quotient groups implying Cartesian product?

[jostpuur]

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

 

[WWGD]

I think if the representation [itex] 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 [itex]G\to H_1[/itex] and [itex]G\to H_2[/itex] given by the assumptions, under the assumption [itex]\#G<\infty[/itex], 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 [itex]\#G=\infty[/itex] remains open.