# Definition:Group Axioms/Left

A group is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions:
 $(\text G 0)$ $:$ Closure Axiom $\displaystyle \forall a, b \in G:$ $\displaystyle a \circ b \in G$ $(\text G 1)$ $:$ Associativity Axiom $\displaystyle \forall a, b, c \in G:$ $\displaystyle a \circ \paren {b \circ c} = \paren {a \circ b} \circ c$ $(\text G_{\text L} 2)$ $:$ Left Identity Axiom $\displaystyle \exists e \in G: \forall a \in G:$ $\displaystyle e \circ a = a$ $(\text G_{\text L} 3)$ $:$ Left Inverse Axiom $\displaystyle \forall a \in G: \exists b \in G:$ $\displaystyle b \circ a = e$