1. ## Isomorphism (Help)

If $H$ and K are normal subgroups of a group $G$ with $HK=G$, prove that $G/(H \cap K) \simeq (G/H) \times (G/K)$.
Hint: If $\phi:G \rightarrow (G/H) \times (G/K)$ is defined by $x \mapsto (xH,xK)$, then $ker \: \phi=H \cap K$; moreover, we have $G=HK$, so that $\displaystyle\cap_{a}aH=HK=\displaystyle\cap_{b}bK$.

I have managed to prove that $ker \: \phi=H \cap K$. I'm having difficulty in proving that $\phi$ is surjective. This is crucial in proving the above isomorphism using 1st Isomorphism Theorem.

2. Originally Posted by Markeur
If $H$ and K are normal subgroups of a group $G$ with $HK=G$, prove that $G/(H \cap K) \simeq (G/H) \times (G/K)$.
Hint: If $\phi:G \rightarrow (G/H) \times (G/K)$ is defined by $x \mapsto (xH,xK)$, then $ker \: \phi=H \cap K$; moreover, we have $G=HK$, so that $\displaystyle\cap_{a}aH=HK=\displaystyle\cap_{b}bK$.

I have managed to prove that $ker \: \phi=H \cap K$. I'm having difficulty in proving that $\phi$ is surjective. This is crucial in proving the above isomorphism using 1st Isomorphism Theorem.
Let $\left(yH,zK\right)$ be in the codomain. Since $HK=G$ what can you say about $y,z$?

3. Hey,

The proof is as below:

We want to show that $\forall (yH,zK) \in (G/H) \times (G/K)$, $\exists g \in G$ such that $\phi(g)=(gH,gK)$.

Since $(yH,zK) \in (G/H) \times (G/K)$, $yH \in G/H$ and $zK \in G/K$.
Since $G=HK$, $yH \in HK/H$ and $zK \in HK/K$.
This implies that $y,z \in HK$.
Since $G=HK$, $g \in HK$.
This implies that $y=g$ and $z=g \: \forall y,z \in G$.
Hence $\exists g \in G$ such that $\phi(g)=(gH,gK)$.
Therefore $(G/H) \times (G/K) \subseteq Im \; \phi$.
Clearly, $Im \; \phi \subseteq (G/H) \times (G/K)$.
Finally, we conclude that $(G/H) \times (G/K)=Im \; \phi$.

From there, we can establish the isomorphism using First Isomorphism Theorem.

So, is the above proof correct?