# Category:Normality in Groups

This category contains results about Normality in Groups in the context of Group Theory.
Definitions specific to this category can be found in Definitions/Normality in Groups.

Let $G$ be a group.

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

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

### Definition 1

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

### Definition 2

Every right coset of $N$ in $G$ is a left coset

that is:

The right coset space of $N$ in $G$ equals its left coset space.

### Definition 3

$\forall g \in G: g \circ N \circ g^{-1} \subseteq N$
$\forall g \in G: g^{-1} \circ N \circ g \subseteq N$

### Definition 4

$\forall g \in G: N \subseteq g \circ N \circ g^{-1}$
$\forall g \in G: N \subseteq g^{-1} \circ N \circ g$

### Definition 5

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

### Definition 6

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

### Definition 7

$N$ is a normal subset of $G$.

## Subcategories

This category has the following 7 subcategories, out of 7 total.