Unfolding all definitions, a group $(G,\ast)$ fulfills the following properties:

# Axiom: Axioms of Group

1. closure: $x \ast y\in G$ for all $x,y\in G$.
2. associativity: $$(x\ast y)\ast z=x\ast (y\ast z)\,$$.
3. existence of neutral element: There is an element $e\in G$ with $e\ast x=x\ast e=x$ for all $x\in G$.
4. existence of inverse: 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.

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### References

#### Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001
2. Lang, Serge: "Algebra - Graduate Texts in Mathematics", Springer, 2002, 3rd Edition