# Category:Normal Subgroups

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This category contains results about Normal Subgroups.
Definitions specific to this category can be found in Definitions/Normal Subgroups.

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

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

### Definition 4

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

### Definition 5

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

### Definition 6

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

### Definition 7

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