Does there exist a group $G$ with a proper subgroup $K$ in which for all $a, b \in G - K$ such that $ab \neq e$, we have $ab \in G - K$?

Question

I am currently studying group theory and I asked myself various questions, though I was able to solve almost all of them, I could not answer the following one:

Does there exist a group $G$ with a proper subgroup $K$ in which for all $a, b \in G - K$ such that $ab \neq e$, we have $ab \in G - K$?

Please help me, I am completely stuck!

Answer 1

This can only hold if $K$ is trivial. Indeed, if $K$ is nontrivial, $k\in K$ with $k\neq e$, and $a\notin K$, then $a,a^{-1}k$ are both not in $K$, but $e\neq k = a(a^{-1}k) \in K$.

And if $K$ is trivial, this necessarily holds.