◀ ▲ ▶Branches / Algebra / Definition: Commutative (Abelian) Group
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 $x\ast x^{1} =x^{1}\ast x=e.$
Mentioned in:
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
 Knauer Ulrich: "Diskrete Strukturen  kurz gefasst", Spektrum Akademischer Verlag, 2001
Adapted from CC BYSA 3.0 Sources:
 Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück