# Definition:Abelian Group Axioms

An algebraic structure $\left({G, +}\right)$ is an abelian group iff the following conditions are satisfied:
 $(G0)$ $:$ Closure $\displaystyle \forall x, y \in G:$ $\displaystyle x + y \in G$ $(G1)$ $:$ Associativity $\displaystyle \forall x, y, z \in G:$ $\displaystyle x + \left({y + z}\right) = \left({x + y}\right) + z$ $(G2)$ $:$ Identity $\displaystyle \exists 0 \in G: \forall x \in G:$ $\displaystyle 0 + x = x = x + 0$ $(G3)$ $:$ Inverse $\displaystyle \forall x \in G: \exists -x \in G:$ $\displaystyle x + \left({-x}\right) = 0 = \left({-x}\right) + x$ $(C)$ $:$ Commutativity $\displaystyle \forall x, y \in G:$ $\displaystyle x + y = y + x$