# Category:Abelian Groups

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

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

An **abelian group** is a group $G$ where:

- $\forall a, b \in G: a b = b a$

That is, every element in $G$ commutes with every other element in $G$.

## Subcategories

This category has the following 21 subcategories, out of 21 total.

### A

### B

- Boolean Group is Abelian (3 P)

### C

- Category of Abelian Groups (1 P)
- Cauchy's Group Theorem (4 P)
- Cyclic Group is Abelian (3 P)

### E

### G

- Graded Abelian Groups (empty)

### P

### S

## Pages in category "Abelian Groups"

The following 78 pages are in this category, out of 78 total.

### A

- Abelian Group Factored by Prime
- Abelian Group Factored by Prime/Corollary
- Abelian Group Induces Commutative B-Algebra
- Abelian Group Induces Entropic Structure
- Abelian Group is Simple iff Prime
- Abelian Group of Order Twice Odd has Exactly One Order 2 Element
- Abelian Group of Prime-power Order is Product of Cyclic Groups
- Abelian Group of Prime-power Order is Product of Cyclic Groups/Corollary
- Abelian Group of Semiprime Order is Cyclic
- Abelianization of Group is Abelian
- Additive Group of Integers is Countably Infinite Abelian Group
- All Elements Self-Inverse then Abelian

### C

- Cauchy's Group Theorem
- Center of Group is Abelian Subgroup
- Commutators are Identity iff Group is Abelian
- Condition for Algebra Loop to be Abelian Group
- Correspondence between Abelian Groups and Z-Modules
- Correspondence between Abelian Groups and Z-Modules/Homomorphisms
- Cosets in Abelian Group
- Cyclic Group is Abelian

### E

### F

### G

### I

- Induced Group Product is Homomorphism iff Commutative/Corollary
- Inverse Completion of Commutative Semigroup is Abelian Group
- Inverse Mapping in Induced Structure of Homomorphism to Abelian Group
- Inversion Mapping is Automorphism iff Group is Abelian
- Inversion Mapping is Isomorphism from Ordered Abelian Group to its Dual
- Isomorphism of Abelian Groups

### M

### O

### P

### Q

### S

- Sequence of Integers defining Abelian Group
- Set of Homomorphisms to Abelian Group is Subgroup of All Mappings
- Set of Linear Transformations under Pointwise Addition forms Abelian Group
- Set of Subgroups of Abelian Group form Subsemigroup of Power Structure
- Set System Closed under Symmetric Difference is Abelian Group
- Structure Induced by Abelian Group Operation is Abelian Group
- Subgroup Generated by Commuting Elements is Abelian
- Subgroup of Abelian Group is Abelian
- Subgroup of Abelian Group is Normal
- Subgroup of Elements whose Order Divides Integer
- Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup
- Subset Product of Abelian Subgroups
- Subsets Greater Than and Less Than Identity of Ordered Abelian Group are Isomorphic Ordered Semigroups
- Symmetric Difference on Power Set forms Abelian Group