# Definition:Normal Subgroup/Definition 5

## Definition

Let $G$ be a group.

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

$N$ is a normal subgroup of $G$ if and only if:

 $\ds \forall g \in G: \,$ $\ds N$ $=$ $\ds g \circ N \circ g^{-1}$ $\ds \forall g \in G: \,$ $\ds N$ $=$ $\ds g^{-1} \circ N \circ g$

where $g \circ N$ etc. denotes the subset product of $g$ with $N$.

That is:

$\forall g \in G: \map {\kappa_g} N = N$

where $\map {\kappa_g} N$ denotes the inner automorphism of $N$ by $g$.

## Notation

The statement that $N$ is a normal subgroup of $G$ is represented symbolically as $N \lhd G$.

A normal subgroup is often represented by the letter $N$, as opposed to $H$ (which is used for a general subgroup which may or may not be normal).

To use the notation introduced in the definition of the conjugate:

$N \lhd G \iff \forall g \in G: N^g = N$

## 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:

 $\ds \forall g \in G: \,$ $\ds \leftparen {n \in N}$ $\iff$ $\ds \rightparen {g \circ n \circ g^{-1} \in N}$ $\ds \forall g \in G: \,$ $\ds \leftparen {n \in N}$ $\iff$ $\ds \rightparen {g^{-1} \circ n \circ g \in N}$

which is another way of stating that $N$ is normal if and only if $N$ stays the same under all inner automorphisms of $G$.

Some sources use distinguished subgroup.