Axiom: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 | \(\ds \forall x, y \in G:\) | \(\ds x + y \in G \) | |||||
\((\text G 1)\) | $:$ | 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 | \(\ds \forall x, y \in G:\) | \(\ds x + y = y + x \) |