Group theory/Finite/Order/Different/Fact

Let ${}G$ be a finite group.

Then every element

${}g\in G$ has finite

order.

The powers

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

are all different.

Write a proof