abelian group

Abelian group

A group $(A,\circ )$ is abelian iff:
$\circ$ is associative
$A$ has an neutral element
every element has an inverse element
and $\circ$ is commutative.

The group operation in abelian groups is usually denoted with $+$.

Example: $(\mathbb{Z},+)$

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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