# 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 $G$.

See Inner Automorphism Maps Subgroup to Itself iff Normal.

Some sources use **distinguished subgroup**.

## Also see

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: Morphisms - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 46$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $8 \ \text{(i)}$