Definition: Cyclic Group, Order of an Element

A group \((G,\ast)\) is called cyclic group, if there exists an element \(g\in G\) with \(G=\{g^n|~n\in\mathbb Z\}\), i.e. all elements of \(G\) can be represented by positive powers of an element \(g\). In this case, we write \(G=\langle g \rangle,\) i.e. \(G\) is generated by by the element \(g\in G\).

The order of an element $a\in G$ is the smallest natural number $n\in\mathbb N,$ for which $a^n=e$ where $e\in G$ is the neutral element of the group. It is denoted by $\operatorname{ord}(a):=n.$ If such a natural number $n$ does not exist, we write $\operatorname{ord}(a)=\infty.$

Examples: 1
Explanations: 2
Lemmas: 3 4
Proofs: 5 6 7 8 9 10 11 12
Propositions: 13 14 15
Theorems: 16 17

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013
  2. Lang, Serge: "Algebra - Graduate Texts in Mathematics", Springer, 2002, 3rd Edition