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

## Definition: Commutative (Abelian) Group

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]

### CC BY-SA 3.0

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