Abelian group

Saying that \(G\) is abelian adds no new element or operation: it is a condition on every pair of elements. For all \(a,b\in G\) we have \(a\ast b=b\ast a\). Associativity, identity and inverses are already part of every group; commutativity only adds that the order of the factors does not change the product. The name honors Niels Henrik Abel.

The closest example is numbers under addition. In \((\mathbb{Z},+)\) we have \(3+5=5+3\), and the same holds for any pair of integers, so \((\mathbb{Z},+)\) is abelian; so are \((\mathbb{R},+)\) and matrices of the same size under addition. The symmetric group \(S_3\), on the other hand, is not: composing from right to left, \((1\,2)(1\,3)=(1\,3\,2)\) but \((1\,3)(1\,2)=(1\,2\,3)\). A single pair that does not commute is enough to say the group is not abelian.

That is why proving and refuting are different tasks. To prove that a group is abelian you must give an argument valid for every pair \(a,b\): checking specific examples, however many, is not enough. To refute it, a single counterexample suffices. It is common to write the operation of an abelian group as addition, with identity \(0\) and inverse \(-a\), but that is only a notation convention.

Do not confuse commuting with associating: the equality \((a\ast b)\ast c=a\ast(b\ast c)\) is associativity and every group has it. In an abelian group many computations simplify; for example, \((a\ast b)^{n}=a^{n}\ast b^{n}\) and every subgroup is normal. Moreover, finite abelian groups are completely classified, which is not the case for groups in general.