# Definition:Group Axioms/Right

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## Definition

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

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