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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:

Notes

| | | | | created: 2014-06-08 22:47:56 | modified: 2020-06-27 08:16:19 | by: bookofproofs | references: [577], [6907]

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

Bibliography (further reading)

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