# Category:Normal Subgroups

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This category contains results about 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

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

### 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 17 subcategories, out of 17 total.

### A

### C

- Central Subgroup is Normal (3 P)
- Conjugacy Class Equation (3 P)

### E

- Examples of Normal Subgroups (26 P)

### I

- Internal Direct Product Theorem (11 P)

### N

### Q

### S

## Pages in category "Normal Subgroups"

The following 89 pages are in this category, out of 89 total.

### C

- Center of Group is Normal Subgroup
- Center of Group of Order Prime Cubed
- Central Subgroup is Normal
- Centralizer is Normal Subgroup of Normalizer
- Characteristic Subgroup of Normal Subgroup is Normal
- Chinese Remainder Theorem (Groups)
- Complement of Normal Subgroup is Isomorphic to Quotient Group
- Condition for Subgroup of Monoid to be Normal
- Condition for Subgroup of Power Set of Group to be Quotient Group
- Condition on Congruence Relations for Cancellable Monoid to be Group
- Congruence Modulo Normal Subgroup is Congruence Relation
- Congruence Relation induces Normal Subgroup
- Congruence Relation on Group induces Normal Subgroup
- Conjugacy Class Equation
- Conjugacy Class of Element of Center is Singleton/Corollary
- Correspondence Theorem (Group Theory)

### F

### G

### I

- Image of Canonical Injection is Normal Subgroup
- Inner Automorphism Maps Subgroup to Itself iff Normal
- Inner Automorphisms form Normal Subgroup of Automorphism Group
- Internal Direct Product Theorem
- Internal Group Direct Product Isomorphism
- Internal Group Direct Product of Normal Subgroups
- Intersection of Abelian Subgroups is Normal Subgroup of Subgroup Generated by those Subgroups
- Intersection of Normal Subgroup with Center in p-Group
- Intersection of Normal Subgroup with Sylow P-Subgroup
- Intersection of Normal Subgroups is Normal
- Intersection with Normal Subgroup is Normal

### K

### N

- Normal Subgroup iff Normalizer is Group
- Normal Subgroup induced by Congruence Relation defines that Congruence
- Normal Subgroup is Kernel of Group Homomorphism
- Normal Subgroup of p-Group of Order p is Subset of Center
- Normal Subgroup of Subset Product of Subgroups
- Normal Subgroup Test
- Normal Subgroup which is Subset of Normal Subgroup is Normal in Subgroup
- Normality Relation is not Transitive

### P

### Q

### S

- Second Isomorphism Theorem for Groups
- Second Isomorphism Theorem/Groups
- Set of Normal Subgroups of Group is Subsemigroup of Power Set Semigroup
- Set of Normal Subgroups of Group is Subsemigroup of Power Set under Intersection
- Smallest Normal Subgroup containing Set
- Stabilizer is Normal iff Stabilizer of Each Element of Orbit
- Subgroup Containing all Squares of Group Elements is Normal
- Subgroup Containing all Squares of Group Elements is Normal/Corollary
- Subgroup equals Conjugate iff Normal
- Subgroup is Normal iff Contains Conjugate Elements
- Subgroup is Normal iff it contains Product of Inverses
- Subgroup is Normal iff Left Coset Space is Right Coset Space
- Subgroup is Normal iff Left Cosets are Right Cosets
- Subgroup is Normal iff Normal Subset
- Subgroup is Normal Subgroup of Normalizer
- Subgroup is Subset of Conjugate iff Normal
- Subgroup is Superset of Conjugate iff Normal
- Subgroup of Abelian Group is Normal
- Subgroup of Index 2 is Normal
- Subset has 2 Conjugates then Normal Subgroup
- Subset Product of Normal Subgroups is Normal
- Subset Product of Normal Subgroups with Trivial Intersection
- Subset Product with Normal Subgroup as Generator
- Subset Product with Normal Subgroup is Subgroup
- Subset Products of Normal Subgroup with Normal Subgroup of Subgroup
- Sylow p-Subgroup is Unique iff Normal
- Symmetric Group has Non-Normal Subgroup