Normal subgroups imply left cosets equal to right cosets

up:: Normal Subgroup

Let $G$ be a Group and $N⊴G$ a Normal Subgroup.

Let $g\in G$. We seek to prove that $gN=Ng$. It follows trivially from “summing zeros”.

1) $gN\subseteq Ng$

Let $gn\in gN$. Then

$$gn=gn(g^{-1}g=\underset{\in N}{\underbrace{(gn{g}^{-1})}}g\in Ng$$

by the definition of normal subgroup.

2) $Ng\subseteq gN$

Let $ng\in Ng$. Then, similarly,

$$ng=(g{g}^{-1})ng=g\underset{\in N}{\underbrace{(g^{-1}ng)}}\in gN$$

by the definition of normal subgroup.

Thus, $gN=Ng$ for normal subgroups.