# 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).

Parents:

• Group

The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.

• Algebraic structure
• I strongly recommend keeping to the standard term “abelian group,” even though “commutative group” would be more systematic and sensible. The term “abelian group” is universal—I don’t know a single mathematician, book, or paper that uses the term “commutative group”—and people comparing what they read here to what they read anywhere else are just going to be confused, and/​or are going to confuse third parties when they ask questions.