> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/400-abstract-algebra/410-group-theory/abelian-group.md).

# 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>

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*.