> [!NOTE] Theorem (Unique identity) > Let $(G,\cdot)$ be a [[Groups|group]]. Then $(G,\cdot)$ has a unique [[Group Identity Element|identity element]]. Proof. Suppose $e,e'\in G$ are identity elements. Then for all $g\in G$ we have $a\cdot e=e\cdot a=a\tag{1}$and $a\cdot e'=e'\cdot a=a \tag{2}$If we let $a=e$ in $(2)$ we have $e\cdot e'=e$But if we let $a=e'$ in $(1)$ we have $e\cdot e'=e'$Thus $e'=e.$