Left cosets equal right cosets implies normal subgroup

up:: Coset

Let $G$ be a Group and $N\le G$ be a Subgroup of $G$.

By hypothesis, left cosets equal right cosets, which means that, for a given $g\in G$, $gN=Ng$.

That is, given $n\in N$, $gn=ng⟹gn{g}^{-1}\in N$.

Thus, $N$ is a Normal Subgroup.