Abelian group

An abelian group is a group \(G=(X, \bullet)\) where \(\bullet\) is commutative. In other words, the group operation satisfies the five axioms:

1. Closure: For all \(x, y\) in \(X\), \(x \bullet y\) is defined and in \(X\). We abbreviate \(x \bullet y\) as \(xy\).

2. Associativity: \(x(yz) = (xy)z\) for all \(x, y, z\) in \(X\).

3. Identity: There is an element \(e\) such that for all \(x\) in \(X\), \(xe=ex=x\).

4. Inverses: For each \(x\) in \(X\) is an element \(x^{-1}\) in \(X\) such that \(xx^{-1}=x^{-1}x=e\).

5. Commutativity: For all \(x, y\) in \(X\), \(xy=yx\).

The first four are the standard group axioms; the fifth is what distinguishes abelian groups from groups.

Commutativity gives us license to re-arrange chains of elements in formulas about commutative groups. For example, if in a commutative group with elements \(\{1, a, a^{-1}, b, b^{-1}, c, c^{-1}, d\}\), we have the claim \(aba^{-1}db^{-1}=d^{-1}\), we can shuffle the elements to get \(aa^{-1}bb^{-1}d=d^{-1}\) and reduce this to the claim \(d=d^{-1}\). This would be invalid for a nonabelian group, because \(aba^{-1}\) doesn’t necessarily equal \(aa^{-1}b\) in general.

Abelian groups are very well-behaved groups, and they are often much easier to deal with than their non-commutative counterparts. For example, every Subgroup of an abelian group is normal, and all finitely generated abelian groups are a direct product of cyclic groups (the structure theorem for finitely generated abelian groups).