Definition:Normal Subgroup

Definition
Let $G$ be a group.

Let $N$ be a subgroup of $G$.

$N$ is a normal subgroup of $G$ iff:

Also known as
It is usual to describe a normal subgroup of $G$ as normal in $G$.

Some sources refer to a normal subgroup as an invariant subgroup or a self-conjugate subgroup.

This arises from Definition 6:

which is another way of stating that $N$ is normal iff $N$ is invariant under all inner automorphisms of $G$.

Also see

 * Equivalence of Definitions of Normal Subgroup


 * Definition:Subnormal Subgroup
 * Definition:Abnormal Subgroup
 * Definition:Weakly Abnormal Subgroup
 * Definition:Contranormal Subgroup
 * Definition:Self-Normalizing Subgroup
 * Definition:Pronormal Subgroup
 * Definition:Weakly Pronormal Subgroup
 * Definition:Paranormal Subgroup
 * Definition:Polynormal Subgroup