# Proposition: Simple Calculations Rules in a Group

In a group $$(G,\ast)$$, the following calculation rules follow immediately from the definition:

1. The inverse element of $$e$$ is $$e$$ itself, symbolically $$e^{-1}=e$$.
2. The equation $$(x^{-1})^{-1}=x$$ holds for all $$x\in G$$.
3. For all $$a,b\in G$$, the equation $$a\ast x=b$$ is uniquely solvable in $$G$$, and the solution is $$x=a^{-1}\ast b$$.
4. For all $a,b\in G$, the equation $$y\ast a=b$$ is uniquely solvable in $G$, and the solution is $y=b\ast a^{-1}.$
5. A group is cancellative, i.e. if $a\ast x=a\ast y$ then $x=y$ and if $x\ast a=y\ast a$ then $x=y.$
6. The equation $$(x\ast y)^{-1}=y^{-1}\ast x^{-1}$$ holds for all $$x,y\in G$$.
7. $$x_1\ast x_2\ast x_3\ast \ldots\ast x_n:=(\ldots((x_1\ast x_2)\ast x_3)\ast \ldots\ast x_n$$ holds for all $$x_i\in G$$.

If the group $$(G,\ast)$$ is Abelian, then it also follows that

# 8. $$x_{k_1}\ast x_{k_2}\ast \ldots\ast x_{k_n}=x_{1}\ast x_{2}\ast \ldots\ast x_{n}$$ holds for any permutation $$k_1,\ldots,k_n$$ of the indices $$1,\ldots, n$$.

Proofs: 1

Explanations: 1
Proofs: 2

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983