Abelian group

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Many common number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers are abelian groups with the group operation being addition. In contrast, symmetry groups and permutation groups, which describe the symmetry of a figure and the ways to rearrange the elements in a list respectively, are often non-commutative. Symmetry groups and permutation groups consist of maps; a group consisting of maps is commutative if and only if the equality ${\displaystyle \scriptstyle f\circ g=g\circ f}$ (it means ${\displaystyle f(g(x))=g(f(x))}$ for all x) holds for all maps f, g in the group.