Definition:Abelian Group Axioms

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Definition

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      \(\displaystyle \forall x, y \in G:\) \(\displaystyle x + y = y + x \)             


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