Group theory/Finite/Order/Different/Fact/Proof/Exercise

Let ${}G$ be a finite group. Show that every element ${}g\in G$ has finite order, and show that the powers

$g^{0}=e_{G},\,g^{1}=g,\,g^{2},\ldots ,g^{\operatorname {ord} \,(g)-1}$

are all different.

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