A group records symmetries, but not every symmetry system behaves like addition. If $\sigma$ and $\tau$ are permutations, applying $\sigma$ and then $\tau$ can give a different result from applying $\tau$ and then $\sigma$. The same obstruction appears in matrix multiplication, rotations of space, and symmetries of polygons. Order matters, so the algebra must remember conjugation, normality, and the difference between left and right actions.

Abelian groups isolate the opposite situation: the operation is still invertible and associative, but order no longer matters. This does not empty the subject. It changes the questions. Instead of tracking conjugacy classes, we ask how elements decompose into independent pieces, how torsion is organised, how generators and relations describe the group, and why every abelian group behaves like a module over $\mathbb{Z}$.

The point of commutativity is not only convenience. It is the reason addition in $\mathbb{Z}$, vector addition, homology groups, divisor class groups, and many algebraic invariants all land in the same category. Abelian groups are often what remains when a complicated object is measured by an additive invariant.

The first contrast is visible before any formal definition. Addition of integers has no memory of order, while composition of permutations does. A small permutation computation shows that the ordinary group axioms cannot force commutativity by themselves, so the abelian condition must be added deliberately.

[example: Order Matters in $S_3$] Let $S_3$ be the group of permutations of $\{1,2,3\}$ under composition, with the convention that $\sigma\tau$ means "apply $\tau$ first, then apply $\sigma$." Put $\sigma=(12)$ and $\tau=(23)$, so $\sigma$ swaps $1$ and $2$ and fixes $3$, while $\tau$ swaps $2$ and $3$ and fixes $1$. For the product $\sigma\tau$, we compute each value: \begin{align*} (\sigma\tau)(1)=\sigma(\tau(1))=\sigma(1)=2. \end{align*} \begin{align*} (\sigma\tau)(2)=\sigma(\tau(2))=\sigma(3)=3. \end{align*} \begin{align*} (\sigma\tau)(3)=\sigma(\tau(3))=\sigma(2)=1. \end{align*} Thus $\sigma\tau$ sends $1\mapsto 2$, $2\mapsto 3$, and $3\mapsto 1$, so $\sigma\tau=(123)$. For the reversed product $\tau\sigma$, the order of application changes: \begin{align*} (\tau\sigma)(1)=\tau(\sigma(1))=\tau(2)=3. \end{align*} \begin{align*} (\tau\sigma)(2)=\tau(\sigma(2))=\tau(1)=1. \end{align*} \begin{align*} (\tau\sigma)(3)=\tau(\sigma(3))=\tau(3)=2. \end{align*} Thus $\tau\sigma$ sends $1\mapsto 3$, $3\mapsto 2$, and $2\mapsto 1$, so $\tau\sigma=(132)$. Since $(123)\ne(132)$, we have $\sigma\tau\ne\tau\sigma$. Therefore the group axioms alone do not force commutativity. [/example]

The failure in $S_3$ is the obstruction abelian groups remove. Once multiplication commutes, every subgroup is normal, every quotient by a subgroup is available, and conjugation ceases to carry information. The price is that we have restricted the world; the reward is a theory with strong classification theorems and a close relationship to linear algebra.

## Definition

The parent notion is a [group](/page/Group): a set with an associative binary operation, an identity element, and inverses. The extra question is whether the operation depends on order. If it does not, the group behaves like addition; if it does, the noncommutative part of group theory enters. This page is the child case in which the parent group structure is already present and commutativity is the additional requirement.

[definition: Abelian Group] An abelian group is a set $G$ equipped with a binary operation \begin{align*} \cdot:G\times G\to G \end{align*} such that $(G,\cdot)$ is a group and \begin{align*} gh = hg \end{align*} for all $g,h \in G$. [/definition]

The word "abelian" is therefore a property of a group, not a replacement for the group axioms. Every abelian group has an identity element, inverses, and associativity before commutativity is even mentioned. The rest of the chapter studies what this added symmetry buys: additive notation, stable quotients, cyclic decomposition, torsion, and the largest abelian quotient of a general group.

## Notation and Commutators

Because many abelian examples behave like addition, we need a notation that exposes sums, zero, negatives, and integer multiples rather than hiding them behind multiplication. This is not a new object; it is a change of language that matches the commutative situation.

[definition: Additive Notation for an Abelian Group] Let $G$ be an abelian group. Additive notation writes the group operation as $+$, the identity element as $0$, and the inverse of $g \in G$ as $-g$. For a positive integer $n$ and $g \in G$, write \begin{align*} ng = \underbrace{g+\cdots+g}_{n\text{ times}}. \end{align*} Also write $0g=0$ and $(-n)g=-(ng)$. [/definition]

In elementary group theory, additive notation is most natural for abelian groups because $g+h=h+g$ resembles ordinary addition. To compare abelian and nonabelian groups, however, we also need multiplicative notation for the failure of two elements to commute. The expression should vanish exactly when $gh$ and $hg$ agree, and this leads to the commutator.

[definition: Commutator] Let $G$ be a group. The commutator map is the function \begin{align*} [-,-]:G\times G\to G \end{align*} defined by \begin{align*} [g,h] = ghg^{-1}h^{-1}. \end{align*} [/definition]

The commutator is designed so that $[g,h]=e$ precisely when $gh=hg$. A definition of abelian group asks for commutativity directly, but in practice we often need a test that works inside a larger nonabelian group and can be converted into relations. The next theorem supplies that test: it turns the global phrase "all elements commute" into the vanishing of a family of explicit elements.

[quotetheorem:9535]

This criterion says that abelian groups are the groups whose internal conjugation noise has disappeared. It also gives a practical way to prove noncommutativity: find a single pair whose commutator is not the identity. The same idea will reappear when we build the largest abelian quotient of a group.

## Examples, Non-Examples, and Centrality

### Additive Models