> [!NOTE] Lemma (Order of identity is necessarily $1$) > Let $G$ be a [[Groups|group]] and $g\in G.$ Then $g$ has [[Order of Group Element|order]] $1$ iff $g$ is the [[Group Identity Element|identity element]]. ^b169c8 **Proof**: ($\implies$) Suppose has order $1.$ Then $g^{1}=1.$ Thus $g=1$ is the identity element. ($\impliedby$) Conversely, the identity element clearly has order $1.$