An abelian (or [[Commutative Property|commutative]]) [[Group]] is any [[Group]] $[A, \cdot]$ whose [[Binary Operation|operation]] $*$ is [[Commutative Property|commutative]] on elements of $G$. These are the [[Axiom|Axioms]] that an [[Abelian Group|abelian group]] must satisfy (where $a,b,c,e\in A$ and $e=\op{id}_{A}$): $\huge \begin{align} a*(b* c) &= (a * b) * c \\ a * e &= a\\ a * a^{-1} &= e \\ a * b &= b * c \end{align}$ A way to see if a [[Finite Group]] is an [[Abelian Group]] is if its [[Cayley Table]] is symmetric over the diagonal.