Normal Subgroup

up:: Subgroup (Mathematics)

Given a Group $G$ and a Subgroup $N\le G$, $N$ is a normal subgroup (denoted $N⊴G$) if

$$\forall g\in G,\forall n\in N:g^{-1}ng\in N$$

Equivalent definition

Normal subgroups imply left cosets equal to right cosets.
This is equivalent to requiring

$$\forall g\in G,\forall n\in N:ng=gn⟺\forall g\in G:gN=Ng$$

that is, the Coset of $N$, $gN$, are equal to its right cosets $Ng$.

Equivalence to a homomorphism’s kernel

Given a Group Homomorphism $\phi :G\to G^{\prime }$, then A homomorphism’s kernel is a normal subgroup.

The reciprocal is also true: a normal subgroup $N⊴G$ induces a homomorphism

$$\begin{array}{rl}\phi :& G\to G/N\\ & g\mapsto gN\end{array}$$

whose kernel is $\ker \phi =N$.