This is a problem from the book "Berkeley Problems in Mathematics":
Let $G$ be a group of order $120$, let $H$ be a subgroup of order $24$, and assume that there is at least one coset of $H$ (other than $H$ itself) which is equal to some right coset of $H$. Prove that $H$ is a normal subgroup of $G$.
To be honest, I couldn't make much progress. I would appreciate any kind of help. Thanks.