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## Definition: Group

A group $(G,\ast)$ is a monoid, in which an inverse element exists for each element, i.e. for all $x\in G$ there is an $x^{-1}\in G$ with $x\ast x^{-1} =x^{-1}\ast x=e,$ where $e\in G$ is the neutral element.

Please note that $e\in X$ is unique in $G$ and $x^{-1}$ is unique for all $x\in G.$

“Unfolding” all definitions, a group fulfills the following axioms:

• Associativity: $x\ast (y\ast z)=(x\ast y)\ast z$ for all $x,y,z\in G.$
• 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 $x\ast x^{-1} =x^{-1}\ast x=e.$

### Notes

• For technical reasons, these axioms are not minimal.
• It is also possible to define a group if we require only the existence of a left-neutral (respectively a right-neutral), and the existence of left-inverse (respectively a right-inverse) elements.
• The reader might encounter this approach in some sources.

### CC BY-SA 4.0

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