Axiom:Abelian Group Axioms

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 $

\((\text G 0)\)	$:$	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	\(\ds \forall x, y \in G:\)	\(\ds x + y = y + x \)