group

Group

$(A,\circ )$ is a group iff
$\circ$ is associative,
$A$ has an neutral element,
every element has an inverse element.

Example: $({\text{bijective functions}}_A,\circ )$, with $|A|\le 2$

Ordnung/order = cardinality of a group

Potentiation

In a group: $(G,\circ )$ with neutral element $e$, and for $a\in G,n\in \mathbb{Z}$, potentiation is defined as:

$$\begin{array}{r}{a}^n:=\{\begin{array}{l}en=0\\ an=1\\ a^{n-1}\circ an>1\\ (a^{-1}{)}^{-n}n<0\end{array}\end{array}$$

Note: It’s ${}^{-n}$ to turn the negative $n$ into a positive number, e.g. $a^{-3}=(a^{-1}{)}^3$, making the definition recursive for negative exponents.

Theorem: $a^{n+m}=a^n\circ a^m,a^{nm}=(a^n{)}^m$, $\forall n,m\in \mathbb{Z}$

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