What is an Abelian group

Specifically, a set $G$ with a binary operation $\cdot$ forms an Abelian group if it satisfies the following five axioms:

• Closure: For all $a, b \in G$, $a \cdot b \in G$.
• Associativity: For all $a, b, c \in G$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
• Identity element: There exists an element $e \in G$ such that for all $a \in G$, $a \cdot e = e \cdot a = a$.
• Inverse element: For each $a \in G$, there exists an element $a^{-1} \in G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$.
• Commutativity: For all $a, b \in G$, $a \cdot b = b \cdot a$.

The last axiom, commutativity, is what distinguishes an Abelian group from a non-Abelian group.