Relationship between direct sum representation and quotient in Groups Let $V$ is a finite dimensional vector space, and $H_{1}, H_{2}$ be subspaces of $V$ such that $V=H_{1}\oplus H_{2}$. Now $V/H_{1}$ is isomorphic to $H_{2}$.
If we replace the vector space $V$ by a group $G$ and consider $H_1$ ,a normal subgroup of $G$, then under what conditions can we say that if $G=H_{1}\oplus H_{2}$ (which is same as $G = H_1 H_2)$, for some subgroup $H_2$ in $G$, then $G/H_{1}$ is isomorphic to $H_{2}$?
Any help will be appreciated!
 A: 
Lemma. If $H_1, H_2 \subseteq G$ are two subgroups, with $H_1$ normal, such that $H_1 \cap H_2 = 0$ and $H_1 H_2 = G$, then $G/H_1 \cong H_2$.

Note that when $H_1$ is normal, the set $H_1 H_2$ is just the set of products $h_1 h_2$ with $h_1 \in H_1$ and $h_2 \in H_2$ (the statement here is that the latter subset is already a subgroup).
Proof. Consider the quotient map $G \to G/H_1$, and restrict it to a map $\pi \colon H_2 \to G/H_1$. Then the assumptions imply that $\pi$ is injective and surjective (think!).
A: Different proof of Remy's Lemma:


*

*Observe $g \in G$ is uniquely $g=ab$ for some $(a,b) \in H_1 \times H_2$

*Define the projection map onto $H_2$ $\phi:G \to H_2$, $\phi_2(g=ab):=b$.

*$\phi_2$ is a surjective group homomorphism with $\ker \phi_2 = H_1$.

*By (3), apply the 1st isomorphism theorem.

Edit: Proof $\phi_2$ is a group homomorphism (wow that was hard): Let $abcd \in G$ for $a,c \in H_1$ and $b,d \in H_2$. We must show $\phi_2(abcd)=\phi_2(ab)\phi_2(cd)=bd$. This is true if $abcd=ebd$ for some $e \in H_1$. Choose $e=abcb^{-1}$, which we can do because $H_1$'s normality gives us $bcb^{-1} \in H_1$.
