An important subtype of groups are *commutative* groups. Therefore, they deserve a separate definition.

A **commutative group** \((G,\ast)\) is a group, in which the binary operation \(\ast\) is commutative, i.e. \(x\ast y=y\ast x\) for all \(x,y\in G\).

A commutative group is also called **Abelian**, named after Niels Hendrik Abel (1802 – 1829).

“Unfolding” all definitions, an *Abelian group* fulfills the following axioms:

- Associativity: $x\ast (y\ast z)=(x\ast y)\ast z$ for all $x,y,z\in G.$
- Commutativity: $x\ast y=y\ast x$ for all $x,y\in G.$
- Neutral Element: There is an element $e\in G$ with $e\ast x=x\ast e=x$ for all $x\in G.$
- Inverse elements: For all $x\in G$ there exists an $x^{-1}\in G$ with $x\ast x^{-1} =x^{-1}\ast x=e.$

| | | | | created: 2014-03-22 21:38:36 | modified: 2019-08-04 08:00:41 | by: *bookofproofs* | references: [577], [6907]

[6907] **Brenner, Prof. Dr. rer. nat., Holger**: “Various courses at the University of Osnabrück”, https://de.wikiversity.org/wiki/Wikiversity:Hochschulprogramm, 2014

[577] **Knauer Ulrich**: “Diskrete Strukturen – kurz gefasst”, Spektrum Akademischer Verlag, 2001