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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]