# 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:

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

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

## Subcategories

This category has the following 6 subcategories, out of 6 total.

### C

### G

### I

### P

### Q

## Pages in category "Quotient Groups"

The following 48 pages are in this category, out of 48 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 Epimorphism is Epimorphism
- Quotient Group is Abelian iff All Commutators in Divisor
- Quotient Group is Group
- Quotient Group is Group/Corollary
- Quotient Group of Abelian Group is Abelian
- Quotient Group of Cyclic Group
- Quotient Group of Direct Products
- Quotient Group of General Linear Group by Special Linear Group
- Quotient Group of Ideal is Coset Space
- Quotient Group of Infinite Cyclic Group by Subgroup
- Quotient Group of Integers by Multiples
- Quotient Group of Quotient Group is Isomorphic to Quotient Group by Preimage under Quotient Mapping
- Quotient Group of Reals by Integers is Circle Group
- Quotient Group of Solvable Group is Solvable
- Quotient of Group by Center Cyclic implies Abelian
- Quotient of Group by Itself
- Quotient of Sylow P-Subgroup
- Quotient of Transformation Group acts Effectively
- Quotient Subgroup of Semigroup Induced on Power Set
- Quotient Theorem for Group Epimorphisms
- Quotient Theorem for Group Homomorphisms