Abelian group

An Abelian group is a group that is commutative – in other words, in which the result of multiplying one member of the group by another is independent of the order of multiplication (i.e., AB = BA, where A, B are any two elements in the group). Abelian groups are a generalisation the arithmetic of addition of integers.


Abelian groups, named after Niels Abel, are of central importance in modern mathematics, most notably in algebraic topology. Examples of Abelian groups include the real numbers (under addition), the non-zero real numbers (under multiplication), and all cyclic groups, such as the integers (under addition). Compare with non-Abelian.