## A criterion for the equality of a left and right coset

Let $G$ be a group, $g \in G$, and $N \leq G$. Prove that $gN = Ng$ if and only if $g \in N_G(N)$.

$(\Rightarrow)$ Suppose $gN = Ng$. Right multiplying by $g^{-1}$, we have $gNg^{-1} = Ngg^{-1} = N$, so that $g \in N_G(N)$. $(\Leftarrow)$ Suppose $g \in N_G(N)$. Then $gNg^{-1} = N$, and right multiplying by $g$ we have $gN = Ng$.