Group Theory: let $G$ be a group and let $G=H\times K$, is it true that $G/H\cong K$?

Question

I have been asked to prove or disprove the following statement:

Let $G$ be a group that equals to the inner direct product of $H$ and $K$, in symbols, $G=H\times K$, is it true that the factor group of $G$ by $H$ is isomorphic to $K$, or $G/H\cong K$?

I guess the statement is true, but I have missing pieces in my proof. Define the mapping $\phi: G/H\rightarrow K$ as $\phi(aH)=a$, $a\in K$. If the mapping is a well defined function, then I can readily show that it is a bijection and preserves operation, and hence the isomorphism. So the crux here is proving its well-definedness, that is, proving the codomain of $\phi$ is indeed $K$. I separated this into two cases:

1. If $aH=H$, then $a\in H$. If we also require $a\in K$, then $a\in H\cap K=\left\{e\right\}$, meaning that $a=e$.
2. If $aH\neq H$, then $a\not\in H$. Since $G=H\times K$ implies $H\trianglelefteq G$, $aH=Ha\leqslant G=HK$. Simplifying, we get $Ha=HK$. Here is the problem, can I just say $\exists k\in K$ with $a=k$?

I am not sure if this is illogical, and even if it is not, there must be steps missing in between. Please help.

Answer 1

Hint:

Let $\phi: G \to K$, $\phi(hk) = k$ (each $x \in G$ can be written as $hk$ for $h \in H$ and $k \in K$).

This map is well-defined and surjective of course, and its kernel is..