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Proposition: Simple Calculations Rules in a Group
In a group \((G,\ast)\), the following calculation rules follow immediately from the definition:
 The inverse element of \(e\) is \(e\) itself, symbolically \(e^{1}=e\).
 The equation \((x^{1})^{1}=x\) holds for all \(x\in G\).
 For all \(a,b\in G\), the equation \(a\ast x=b\) is uniquely solvable in \(G\), and the solution is \(x=a^{1}\ast b\).
 For all $a,b\in G$, the equation \(y\ast a=b\) is uniquely solvable in $G$, and the solution is $y=b\ast a^{1}.$
 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 $x=y.$
 The equation \((x\ast y)^{1}=y^{1}\ast x^{1}\) holds for all \(x,y\in G\).
 \(x_1\ast x_2\ast x_3\ast \ldots\ast x_n:=(\ldots((x_1\ast x_2)\ast x_3)\ast \ldots\ast x_n\) 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 \(1,\ldots, n\).
Table of Contents
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Explanations: 1
Proofs: 2
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983