Normal Subgroup

A subgroup $N$ of a group $G$ is normal (written $N \trianglelefteq G$) if $gNg^{-1} = N$ for every $g \in G$, or equivalently if the left and right cosets of $N$ coincide: $gN = Ng$ for all $g$.

Normal subgroups are exactly the subgroups for which the set of cosets inherits a well-defined group operation, forming the quotient group $G/N$. They are also exactly the kernels of group homomorphisms.

In an abelian group, every subgroup is normal. In non-abelian groups, normality is a special property that must be verified.