A question about direct product of subgroups.

Question

Let $H$ and $K$ be two subgroups of a group $G$. Suppose that $H$ is normal in $G$ and $G/H\simeq K$. My question is when $G\simeq H\times K$?

My guess is if $K$ is normal in $G$ and $G=HK$ then $G\simeq H\times K$. I want to know whether my guess is right. I do not need any proof, any help is appreciated.

Answer 1

We need these conditions:

$1.)$ $G=HK$

$2.)$ $H$ and $K$ normal in $G$.

$3.)$ $H \cap K= \{1\}$

Then $G=HK \cong H \times K$

Answer 2

For finite groups, your guess is correct; not sure about the infinite case. For a proof, let $G = HK$, suppose that $H \unlhd G$, $K \unlhd G$, and that $G/H \cong K$. To show that $G\cong H \times K$, we need only show that $H \cap K =1$. We appeal to the diamond isomorphism theorem, which says in this case that $G/H=HK/H \cong K/H \cap K$. But since $G/H \cong K$, we have $K \cong K/H \cap K$, so by order considerations, we must have $H \cap K = 1$.