# Difference between revisions of "Abelian group"

m (typo) |
(removed stub) |
||

Line 30: | Line 30: | ||

Notable examples of abelian groups include the integers under addition, the real numbers under addition, the integers modulo <math>n</math> under addition, the multiplicative group of integers modulo <math>n</math>, and the additive group of any ring. Many matrix groups are ''not'' abelian because matrix multiplication is associative and not commutative. The smallest finite non-abelian group is the dihedral group of order 6. | Notable examples of abelian groups include the integers under addition, the real numbers under addition, the integers modulo <math>n</math> under addition, the multiplicative group of integers modulo <math>n</math>, and the additive group of any ring. Many matrix groups are ''not'' abelian because matrix multiplication is associative and not commutative. The smallest finite non-abelian group is the dihedral group of order 6. | ||

− | |||

[[Category:Definition]] | [[Category:Definition]] | ||

[[Category:Group theory]] | [[Category:Group theory]] |

## Latest revision as of 18:30, 14 June 2020

An **abelian group** is a group in which the group operation is commutative. They are named after Norwegian mathematician Niels Abel.
For a group to be considered **abelian**, it must meet several requirements.

Closure

For all , and for all operations , .

Associativity

For all and all operations , .

Identity Element

There exists some such that .

Inverse Element

For all , there exists some such that

Commutativity

For all , .

A simple example of an abelian group is under addition. It is simple to show that it meets all the requirements.

Closure

For all .

Associativity

For all .

Identity Element

For all .

Inverse Element

For all .

Commutativity

For all .

Seeing as meets all of these requirements under addition, we can say that is abelian under addition.

## Examples

Notable examples of abelian groups include the integers under addition, the real numbers under addition, the integers modulo under addition, the multiplicative group of integers modulo , and the additive group of any ring. Many matrix groups are *not* abelian because matrix multiplication is associative and not commutative. The smallest finite non-abelian group is the dihedral group of order 6.