Normality via left and right cosets

Statement

Let $G$ be a group and let $N$ be a subgroup of $G$. Then $N$ is normal in $G$ if and only if $gN = Ng$ for each $g \in G$.

Remarks

Here $gN$ is a left coset and $Ng$ is a right coset. The equation $gN = Ng$ is equivalent to $gNg^{-1} = N$, by multiplying on the right by $g^{-1}$. The condition does not say that $gn = ng$ for each $n \in N$; it says only that the two sets are equal.

Explanations

L1 human

For a normal subgroup, it does not matter if you multiply by $g$ on the left or on the right: you get the same coset. Thus there is only one kind of coset, and we can just say "the cosets of $N$". This is what makes the quotient group possible.