Definition:Abelian Group Axioms

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Definition

An algebraic structure $\struct {G, +}$ is an abelian group if and only if the following conditions are satisfied:

\((\text G 0)\)   $:$   Closure      \(\displaystyle \forall x, y \in G:\) \(\displaystyle x + y \in G \)             
\((\text G 1)\)   $:$   Associativity      \(\displaystyle \forall x, y, z \in G:\) \(\displaystyle x + \paren {y + z} = \paren {x + y} + z \)             
\((\text G 2)\)   $:$   Identity      \(\displaystyle \exists 0 \in G: \forall x \in G:\) \(\displaystyle 0 + x = x = x + 0 \)             
\((\text G 3)\)   $:$   Inverse      \(\displaystyle \forall x \in G: \exists \paren {-x}\in G:\) \(\displaystyle x + \paren {-x} = 0 = \paren {-x} + x \)             
\((\text C)\)   $:$   Commutativity      \(\displaystyle \forall x, y \in G:\) \(\displaystyle x + y = y + x \)             


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