# Category:Quotient Groups

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This category contains results about **Quotient Groups**.

Definitions specific to this category can be found in Definitions/Quotient Groups.

Let $G$ be a group.

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

Then the left coset space $G / N$ is a group, where the group operation is defined as the coset product:

- $\paren {a N} \paren {b N} = \paren {a b} N$

$G / N$ is called the **quotient group of $G$ by $N$**.

## Subcategories

This category has the following 11 subcategories, out of 11 total.

## Pages in category "Quotient Groups"

The following 42 pages are in this category, out of 42 total.

### C

### G

### I

### P

### Q

- Quotient Epimorphism is Epimorphism/Group
- Quotient Group by Intersection of Normal Subgroups not necessarily Cyclic if Quotient Groups are
- Quotient Group is Abelian iff All Commutators in Divisor
- Quotient Group is Group
- Quotient Group is Group/Corollary
- Quotient Group is Subgroup of Power Structure of Group
- Quotient Group of Abelian Group is Abelian
- Quotient Group of Cyclic Group
- Quotient Group of Direct Products
- Quotient Group of Ideal is Coset Space
- Quotient Group of Infinite Cyclic Group by Subgroup
- Quotient Group of Solvable Group is Solvable
- Quotient of Group by Center Cyclic implies Abelian
- Quotient of Sylow P-Subgroup
- Quotient of Transformation Group acts Effectively
- Quotient Theorem for Group Epimorphisms
- Quotient Theorem for Group Homomorphisms
- Quotient Theorem for Group Homomorphisms/Corollary 1
- Quotient Theorem for Group Homomorphisms/Corollary 2