Normal subgroup

In group theory, a branch of mathematics, a subgroup ${\displaystyle N}$ of the group ${\displaystyle G}$ is normal in ${\displaystyle G}$ if and only if ${\displaystyle {\frac {1}{gng}}\in N}$ for all ${\displaystyle g\in G}$ and ${\displaystyle n\in N}$.

The usual notation for this relation is ${\displaystyle N\triangleleft G.}$