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Definition: Cyclic Group, Order of an Element

A group \((G,\ast)\) is called cyclic, 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 $\operatorname{ord}(a)=\infty.$

| | | | | created: 2014-08-24 21:46:29 | modified: 2020-10-08 15:33:29 | by: bookofproofs | references: [696], [6735]

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Bibliography (further reading)

[696] Kramer Jürg, von Pippich, Anna-Maria: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013

[6735] Lang, Serge: “Algebra – Graduate Texts in Mathematics”, Springer, 2002, 3rd Edition