Let $H, K \lhd G$, then $G = HK \iff G / (H \cap K) \cong G / H \times G / K$

Question

This is a followup of this question

So, let $ (G,*) $ be a group.

Say $ H,K \lhd G $

If $ G $ is finite, then from the previously asked question we conclude: $ \ G = HK \iff G / (H \cap K) \cong G / H \times G / K$

Now, my question is for infinite groups $ G $, is the above conclusion still valid?

Can anyone please counter or prove it?

Answer 1

There are groups $G$ with $G\cong G\times G$. For such a group take $H=K=\{e\}$. Then $HK=\{e\}$ and $G/H\cap K\cong G/H\times G/K$.

[For $G$ we can take (additively) the sum of countably many copies of $\mathbb{Z}$.]