# Thread: Isomorphism (Help)

1. ## Isomorphism (Help)

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

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

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

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

3. Hey,

The proof is as below:

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

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

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

So, is the above proof correct?

Thanks in advance.