Motivation

Definition

Any subgroup $H \subset G$ such that for all $h \in H$ and $a \in G$,

$aha^{-1} \in H$

That is, it is invariant under conjugation with elements of the parent group

If G is Abelian group then all subgroups are normal.

If the only normal subgroups are $\{1\}$ and $G$, the group is called simple.

Proposition. The Quotient set for a normal subgroup is a group, called the Quotient group

lemma. The Canonical projection for the Quotient group is a Group homomorphism