> [!NOTE] **Lemma** > Let $G$ be a [[Finite Group|finite group]] and $g\in G.$ Then $g$ has [[Order of Group Element|finite order]]. **Proof**. We prove the contrapositive. Suppose $g \in G$ has infinite order. Then $g^{0}, g^{1}, g^{2},\dots$ are elements of $G$ which are distinct from one another (since by [[Criteria for Equality of Powers of Group Element]] if $g^{m} =g^{n}$ iff $m=n$). Then $G$ is infinite.