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 Henrik 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 $x\ast x^{-1} =x^{-1}\ast x=e.$

Chapters: 1 2
Corollaries: 3 4
Definitions: 5 6 7 8
Examples: 9
Explanations: 10 11
Lemmas: 12 13
Motivations: 14
Proofs: 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
Propositions: 34 35 36 37 38 39 40 41 42 43 44

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### References

#### Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001

#### Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück