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