Abelian group

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

Definition

An abelian group is a set $A$ together with a binary operation $\cdot$ (or $+$) that combines any two elements $a$ and $b$ to form another element denoted $a \cdot b$ (or $a + b$). To be an abelian group, the set and operation must satisfy five requirements known as the abelian group axioms:

1. Closure: For all $a, b \in A$, the result of the operation $a \cdot b$ is also in $A$.
2. Associativity: For all $a, b, c \in A$, the equation $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ holds.
3. Identity element: There exists an element $e \in A$ such that for all $a \in A$, $e \cdot a = a \cdot e = a$.
4. Inverse element: For each $a \in A$, there exists an element $b \in A$ such that $a \cdot b = b \cdot a = e$.
5. Commutativity: For all $a, b \in A$, $a \cdot b = b \cdot a$.

A group in which the commutativity axiom does not hold is called a non-abelian group (or non-commutative group). When the group operation is commutative, it is customary to use additive notation (with $+$ as the operation, $0$ as the identity, and $-a$ as the inverse of $a$), especially when the group is considered as a module over a ring. Multiplicative notation (with $\cdot$ or juxtaposition, $1$ or $e$ as the identity, and $a^{-1}$ as the inverse) is also common.

Examples

The most familiar example of an abelian group is the set of integers $\mathbb{Z}$ under the operation of addition. The sum of any two integers is an integer, addition is associative and commutative, zero is the identity element, and every integer $n$ has an additive inverse $-n$. Similarly, the rational numbers $\mathbb{Q}$, real numbers $\mathbb{R}$, and complex numbers $\mathbb{C}$ all form abelian groups under addition. Furthermore, the non-zero elements of $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ form abelian groups under multiplication.

Every cyclic group is abelian. If a group is generated by a single element $g$, then every element in the group can be written as $g^n$ for some integer $n$. Since $g^n \cdot g^m = g^{n+m} = g^{m+n} = g^m \cdot g^n$, the group is commutative. The integers modulo $n$, denoted $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}_n$, form a finite cyclic group under addition modulo $n$.

In contrast, the set of invertible $n \times n$ matrices over a field, known as the general linear group $GL(n)$, forms a non-abelian group under matrix multiplication for $n \ge 2$, because matrix multiplication is generally not commutative. Similarly, the symmetric group $S_n$ of all permutations of a set with $n$ elements is non-abelian for $n \ge 3$.

Properties

Abelian groups possess several simplifying properties that make their structure easier to analyze than that of general groups:

• Subgroups: Every subgroup of an abelian group is a normal subgroup. This is because for any subgroup $H$ of an abelian group $G$ and any $g \in G$, $gH = Hg$ trivially holds due to commutativity. Consequently, the quotient group $G/H$ is always well-defined.
• Homomorphisms: The set of all group homomorphisms from an abelian group $G$ to another abelian group $H$, denoted $\text{Hom}(G, H)$, itself forms an abelian group under pointwise addition. If $f, g: G \to H$ are homomorphisms, their sum $(f+g)(x) = f(x) + g(x)$ is also a homomorphism, a property that fails for non-abelian groups.
• Center: The center of an abelian group is the entire group itself, as every element commutes with every other element.
• Automorphisms: The automorphism group of an abelian group, while not necessarily abelian itself, has a well-understood structure, particularly for finite abelian groups.

Finite Abelian Groups

The structure of finite abelian groups is completely classified by the Fundamental Theorem of Finite Abelian Groups. This theorem states that every finite abelian group is isomorphic to a direct sum (or direct product) of cyclic groups of prime-power order.

Specifically, if $G$ is a finite abelian group of order $n$, then $G$ can be expressed as:
$G \cong \mathbb{Z}_{p_1^{k_1}} \oplus \mathbb{Z}_{p_2^{k_2}} \oplus \cdots \oplus \mathbb{Z}_{p_m^{k_m}}$
where $p_i$ are primes (not necessarily distinct) and the product of the orders $p_1^{k_1} p_2^{k_2} \cdots p_m^{k_m} = n$.

Alternatively, the theorem can be stated using invariant factors: every finite abelian group is isomorphic to a direct sum of cyclic groups $\mathbb{Z}_{d_1} \oplus \mathbb{Z}_{d_2} \oplus \cdots \oplus \mathbb{Z}_{d_k}$, where $d_1 | d_2 | \cdots | d_k$ and $d_1 d_2 \cdots d_k = n$. These two decompositions are unique up to the order of the factors, providing a complete classification of finite abelian groups.

Finitely Generated Abelian Groups

The classification extends to finitely generated abelian groups through the Fundamental Theorem of Finitely Generated Abelian Groups. A finitely generated abelian group $G$ is isomorphic to the direct sum of a free abelian group of finite rank and a finite abelian group.

Mathematically, $G \cong \mathbb{Z}^r \oplus \mathbb{Z}_{d_1} \oplus \mathbb{Z}_{d_2} \oplus \cdots \oplus \mathbb{Z}_{d_k}$, where $r \ge 0$ is the rank (or Betti number) of the group, and $d_1 | d_2 | \cdots | d_k$ are integers greater than 1. The integer $r$ and the sequence of invariant factors $d_1, \dots, d_k$ are uniquely determined by $G$. The subgroup $\mathbb{Z}^r$ is the torsion-free part, and the finite sum of cyclic groups is the torsion subgroup, consisting of all elements of finite order.

Connection to Other Mathematical Structures

Abelian groups serve as the foundational building blocks for many other algebraic structures.

• Modules: An abelian group is precisely a module over the ring of integers $\mathbb{Z}$. Many theorems about abelian groups are special cases of theorems about modules over principal ideal domains.
• Rings and Fields: A ring is an abelian group under its addition operation, equipped with a second operation (multiplication) that is associative and distributes over addition. A field is a commutative ring where the non-zero elements form an abelian group under multiplication.
• Vector Spaces: A vector space over a field $F$ is an abelian group under vector addition, along with scalar multiplication by elements of $F$. Thus, every vector space is an abelian group, and the study of vector spaces can be viewed as the study of modules over fields.
• Topological Groups: When an abelian group is equipped with a topology such that the group operations are continuous, it forms a topological abelian group. Pontryagin duality provides a deep framework for studying locally compact abelian groups, establishing a duality between a group and its character group.

Historical Note

The term "abelian group" is named in honor of the Norwegian mathematician Niels Henrik Abel (1802–1829). Abel introduced the concept of commutativity in the context of his work on the solvability of polynomial equations by radicals. He discovered that if the roots of a polynomial equation can be expressed rationally in terms of each other in a commutative manner, the equation is solvable by radicals. The groups of permutations of the roots that exhibit this commutative property were later termed "abelian groups" by Camille Jordan in his 1870 treatise on group theory, cementing Abel's legacy in abstract algebra.