An D21: Algebraic structure $G = (X, \times)$ is an abelian group if and only if

(1)	\begin{equation} \forall \, x, y \in X : x y \in X \end{equation}
(2)	\begin{equation} \forall \, x, y, z \in X : (x y) z = x (y z) \end{equation}
(3)	\begin{equation} \exists \, 1_G \in X : \forall \, x \in X : 1_G x = x 1_G = x \end{equation}
(4)	\begin{equation} \forall \, x \in X : \exists \, x^{-1} \in X : x^{-1} x = x x^{-1} = 1_G \end{equation}
(5)	\begin{equation} \forall \, x, y \in X : x y = y x \end{equation}