# Axiom:Abelian Group Axioms

An algebraic structure $\struct {G, +}$ is an abelian group if and only if the following conditions are satisfied:
 $(\text G 0)$ $:$ Closure $\ds \forall x, y \in G:$ $\ds x + y \in G$ $(\text G 1)$ $:$ Associativity $\ds \forall x, y, z \in G:$ $\ds x + \paren {y + z} = \paren {x + y} + z$ $(\text G 2)$ $:$ Identity $\ds \exists 0 \in G: \forall x \in G:$ $\ds 0 + x = x = x + 0$ $(\text G 3)$ $:$ Inverse $\ds \forall x \in G: \exists \paren {-x}\in G:$ $\ds x + \paren {-x} = 0 = \paren {-x} + x$ $(\text C)$ $:$ Commutativity $\ds \forall x, y \in G:$ $\ds x + y = y + x$