# Abelian group

[https://en.wikipedia.org/wiki/Abelian\_group](https://en.wikipedia.org/wiki/Abelian_group)

**Abelian group** or **commutative group** is a group where the axiom of _commutativity_ is also satisfied. Abelian groups generalize the arithmetic of addition of integers.

Abelian group: the Set of Satisfied Axioms (SSA)

* CLO Closure (Totality)
* IDE Identity
* INV Invertibility
* ASS Associativity
* COM Commutativity

_Abelian group_ is a _group_ in which the result of applying the _group operation_ to two _group elements_ does not depend on the order in which they are written. That is, the group operation is _commutative_.

With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples.

The concept of an abelian group underlies many _fundamental algebraic structures_, such as _fields_, _rings_, _vector spaces_, and _algebras_.

_The theory of abelian groups_ is generally simpler than that of _non-abelian groups_; _finite abelian groups_ are very well understood and fully classified.

## Non-abelian group

[https://en.wikipedia.org/wiki/Non-abelian\_group](https://en.wikipedia.org/wiki/Non-abelian_group)

A **non-abelian group**, sometimes called a **non-commutative group**, is a group `(G, ∗)` in which there are at least two elements `a` and `b` that are non commutative with respect to the group operation: `a ∗ b ≠ b ∗ a`

∃a,∃b ∈ G . a ∗ b ≠ b ∗ a

This class of groups contrasts with abelian groups which are commutative (all group elements commute).

Both _discrete groups_ and _continuous groups_ may be non-abelian. Most of the interesting _Lie groups_ are non-abelian, and these play an important role in _gauge theory_.
