Does $HK = G$ imply that $KH = G$ when $H,K$ aren't subgroups?

Question

Let $G$ be a finite group and let $H$ and $K$ be two "subsets" of $G$ such that $HK = G$. Does that imply that $KH = G$?

Since $H$ and $K$ are not subgroups we cannot use the formula $o(HK) = \frac{o(H)o(K)}{o(H\cap K)}$.

I believe this is false and I am looking for counterexamples.

Thank you.

Answer 1

Hopefully I didn't make any stupid mistake :)

Let $G=S_3$ and $$H= \{ e, (1,2) \} \\ K=\{ e, (1,3), (1,2,3) \}$$

Then $$HK= S_3$$ but $$(1,3)(1,2)=(1,2,3)=(1,2,3)e$$ showing that $KH$ can have at most 5 distinct elements.