Let $G$ be a group and $H$ be a not normal subgroup of $G$. I need to prove that there are two left cosets of $H$: $g_1H,g_2H$ such that $g_1Hg_2H$ is not a left coset of $H$.
I tried to assume the contrary and get to that $H$ is normal, which will mean a contradiction. I tried just using definitions or to find an homomorphism which I can apply the first isomorphism theorem with maybe. However, it didn't work out for me.
Any help would be appreciated.