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 $a$ $\langle a\rangle=\{a^0,a^1,a^2,\ldots a^{n-1}\}$ has the group order equal to the order of the element $a,$ formally $$|\langle a\rangle|=\operatorname{ord}(a)=n.$$

Proofs: 1

Proofs: 1 2