# Definition:Normal Subgroup/Definition 1

## Definition

Let $G$ be a group.

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

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

- $\forall g \in G: g \circ N = N \circ g$

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

## 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 congugate:

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

## 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:

- $\forall g \in G: \paren {n \in N \iff g \circ n \circ g^{-1} \in N}$
- $\forall g \in G: \paren {n \in N \iff g^{-1} \circ n \circ g \in N}$

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**.

## Also see

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 6.6$. Normal subgroups - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 11$ - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.10$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 5$: Groups $\text{I}$: Subgroups - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 49$. Normal subgroups - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Proposition $7.4 \ \text{(d)}$