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Proposition: Finite Order of an Element Equals Order Of Generated Group
Let $(G,\ast)$ be a group and $a\in G$ be an element with the finite order $\operatorname{ord}(a)=n < \infty.$ The group generated by $a$ $\langle a\rangle=\{a^0,a^1,a^2,\ldots a^{n1}\}$ has the group order equal to the order of the element $a,$ formally $$\langle a\rangle=\operatorname{ord}(a)=n.$$
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References
Bibliography
 Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013