We were recently taught in lecture the definition of the semidirect product:
Definition: A group $G$ is a semidirect product of Subgroups $H,K$ if $H$ is normal and the canonical projection $G \to G/H$ induces a isomorphism $K \to G/H$.
It was later stated without any proof that $G$ is the semidirect product of subgroups $H$ and $K$ if and only if $H$ is normal and $HK = G$ and $ H \cap K = 1$.
I do not see how one can prove that nor where the intuition behind that is. I would be glad if someone can explain it to me.