Axiom:Group Axioms

Definition

A group is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions:

$(\text G 0)$	$:$	Closure	$\ds \forall a, b \in G:$	$\ds a \circ b \in G $
$(\text G 1)$	$:$	Associativity	$\ds \forall a, b, c \in G:$	$\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c $
$(\text G 2)$	$:$	Identity	$\ds \exists e \in G: \forall a \in G:$	$\ds e \circ a = a = a \circ e $
$(\text G 3)$	$:$	Inverse	$\ds \forall a \in G: \exists b \in G:$	$\ds a \circ b = e = b \circ a $

These four stipulations are called the group axioms.

The group axioms are also known as the group postulates, but the latter term is less indicative of the nature of these statements.

The numbering of the group axioms themselves is to a certain extent arbitrary.

For example, some sources do not include $\text G 0$ on the grounds that it is taken for granted that $\circ$ is closed in $G$.

However, in the treatment of more abstract aspects of group theory it is recommended that this axiom be taken into account.

\((\text G 0)\)	$:$	Closure	\(\ds \forall a, b \in G:\)	\(\ds a \circ b \in G \)
\((\text G 1)\)	$:$	Associativity	\(\ds \forall a, b, c \in G:\)	\(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)
\((\text G 2)\)	$:$	Identity	\(\ds \exists e \in G: \forall a \in G:\)	\(\ds e \circ a = a = a \circ e \)
\((\text G 3)\)	$:$	Inverse	\(\ds \forall a \in G: \exists b \in G:\)	\(\ds a \circ b = e = b \circ a \)