A [group](Groups.md) is said to be _Abelian_ if its group operation [commutes](Commutators.md). Thus commutativity is [the Abelian group axiom.](Group%20axioms.md#Abelian%20group%20axiom) [Abelian groups](Abelian%20groups.md) are equivalent to $\mathbb{Z}$-[modules](Module.md). %%The claim of Abelian groups being Z-modules is on page 158 of Algebra Chapter 0.%% #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory