# Group

The symmetries of an equilateral triangle — three rotations and three reflections — form a six-element system in which any two symmetries can be composed and any symmetry can be undone. The integers under addition satisfy the same structural rules, yet they form an infinite, commutative system with no geometric content whatsoever. What makes both of these examples instances of a single concept is the definition of a *group*: a [set](/page/Set) equipped with a binary operation that is associative, possesses an identity element, and admits inverses. This austere set of axioms unifies the study of symmetry, [number theory](/page/Number%20Theory), geometry, and [linear algebra](/page/Linear%20Algebra), and provides the language in which most of modern algebra is written.

The power of the group concept lies not in any individual axiom but in their interplay. Associativity allows unambiguous computation with arbitrarily long products. The identity and inverse axioms guarantee that every equation $ax = b$ has a unique solution $x = a^{-1}b$. From these modest requirements flows an extraordinary theory: [Lagrange's theorem](/theorems/841) constrains which [subgroups](/page/Subgroup) can exist, the [isomorphism theorems](/theorems/842) reveal when two groups are structurally identical despite surface differences, and group actions connect abstract algebra to concrete counting problems. The article develops each of these threads, emphasising at each stage what problems the definitions solve, what the theorems do and do not guarantee, and what techniques carry over from one application to the next.

[motivation] ### Symmetry without a framework Consider the symmetries of an equilateral triangle. There are six: three rotations (by $0^\circ$, $120^\circ$, and $240^\circ$) and three reflections. Any two of these symmetries can be composed to obtain a third, and every symmetry can be undone. But without a formal framework, studying these compositions requires a $6 \times 6$ multiplication table built by case analysis. Worse, when we move to the square (eight symmetries), the regular pentagon (ten symmetries), or a molecule with dozens of symmetries, the ad hoc approach collapses under its own weight. The group axioms extract exactly the properties that make symmetry composition tractable. Once we verify that a set of symmetries satisfies the axioms, the entire machinery of group theory — subgroups, cosets, quotients, actions — becomes available without repeating case analyses. ### Equations and structure A different path to the same abstraction begins with equations. Consider the integers modulo $n$, written $\mathbb{Z}/n\mathbb{Z}$. Addition in $\mathbb{Z}/n\mathbb{Z}$ is associative, $0$ serves as an identity, and every element $a$ has an additive inverse $n - a$. These are precisely the group axioms. The same pattern appears in the nonzero elements of a [finite field](/page/Field) $\mathbb{F}_p$ under multiplication, or in the set of invertible $n \times n$ [matrices](/page/Matrix) under matrix multiplication. In each case, the ability to solve equations — finding $x$ such that $a + x = b$, or $Ax = B$ — depends on the same structural features: associativity plus the existence of an identity and inverses. The group axioms codify this observation. ### The unifying abstraction The definition of a group is deliberately austere. It asks for a set, a binary operation, and three properties. This austerity is a feature: it means that theorems proved from the axioms alone apply simultaneously to $\mathbb{Z}$, to $S_n$, to $\operatorname{GL}_n(\mathbb{R})$, and to the symmetry group of any geometric figure. The abstraction converts a menagerie of special cases into a single theory. [/motivation]

## The Axioms

Any attempt to study algebraic structure must begin with the question: what is the minimal set of rules that makes composition, identity, and inversion behave predictably? Requiring too much — commutativity, for instance — would exclude most interesting examples: the symmetric groups $S_n$ for $n \geq 3$, the matrix groups $\operatorname{GL}_n(\mathbb{R})$, and the symmetry groups of most geometric objects are all non-commutative. Requiring too little — dropping associativity — would make even basic calculations ambiguous, since the expression $abc$ could mean either $(ab)c$ or $a(bc)$, potentially yielding different results. The group axioms represent a careful balance: they are weak enough to encompass an enormous range of structures, yet strong enough to support a deep theory.

[definition:Group] A **group** is a pair $(G, \cdot)$ where $G$ is a non-empty set and \begin{align*} \cdot : G \times G &\to G \\ (a, b) &\mapsto a \cdot b \end{align*} is a binary operation satisfying: 1. **Associativity.** For all $a, b, c \in G$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. 2. **Identity.** There exists an element $e \in G$ such that $e \cdot g = g \cdot e = g$ for all $g \in G$. 3. **Inverses.** For each $g \in G$, there exists $g^{-1} \in G$ such that $g \cdot g^{-1} = g^{-1} \cdot g = e$. [/definition]

A natural first concern is whether the identity and inverses promised by the axioms are uniquely determined. If they were not, the notation "$e$" and "$g^{-1}$" would be ambiguous, and every subsequent result would need to specify *which* identity or *which* inverse is intended. Fortunately, uniqueness holds, and the argument is a prototype of a technique that recurs throughout algebra.

[quotetheorem:764]

The proof is worth internalising. For the identity: if $e_1$ and $e_2$ both satisfy the identity axiom, then $e_1 = e_1 \cdot e_2 = e_2$, since the first equality uses $e_2$ as an identity and the second uses $e_1$. For inverses: if $y$ and $z$ are both inverses of $g$, then $y = y \cdot e = y \cdot (g \cdot z) = (y \cdot g) \cdot z = e \cdot z = z$ by associativity. Note how critically this argument depends on associativity — in a non-associative structure, the step $y \cdot (g \cdot z) = (y \cdot g) \cdot z$ fails, and indeed non-associative systems (such as octonions) can have non-unique inverses in certain senses. The technique — assume two candidates and show they must be equal — will reappear in the proofs that kernel elements are unique, that quotient operations are well-defined, and throughout [ring theory](/page/Ring).

What the theorem does *not* address is whether the axioms could be weakened. If we require only a *left* identity and *left* inverses (that is, $e \cdot g = g$ for all $g$ and for each $g$ there exists $h$ with $h \cdot g = e$), then $e$ is also a right identity and $h$ is also a right inverse — the proof is a short exercise in the axioms. However, requiring a left identity and *right* inverses is not sufficient; the resulting structure need not be a group.

When the operation is understood, we write $G$ in place of $(G, \cdot)$ and suppress the dot, writing $ab$ for $a \cdot b$. For abelian groups, the operation is often written as $+$ and the identity as $0$.

### Why commutativity is special

In many of the most familiar algebraic systems — the integers, the rationals, the reals — the operation is commutative: $a + b = b + a$, or $ab = ba$. It is tempting to take this for granted, but commutativity is a genuinely restrictive condition. The symmetric group $S_3$ fails it, as do all matrix groups $\operatorname{GL}_n(\mathbb{R})$ for $n \geq 2$. The groups that do satisfy it have significantly more tractable structure: every subgroup of an abelian group is normal, quotient constructions are simpler, and the [fundamental theorem of finite abelian groups](/theorems/841) provides a complete classification. Isolating this property as a named condition allows the theory to branch: results that require commutativity are cleanly separated from those that do not.

[definition:Abelian Group] A group $G$ is **abelian** if $ab = ba$ for all $a, b \in G$. [/definition]

The integers $(\mathbb{Z}, +)$, the rationals $(\mathbb{Q}, +)$, and the multiplicative group $(\mathbb{R}^{\times}, \cdot)$ of nonzero reals are all abelian. The symmetric group $S_n$ for $n \geq 3$ is the first standard example of a non-abelian group: the transpositions $(1\;2)$ and $(1\;3)$ do not commute.

## Fundamental Examples

The group axioms are satisfied by an enormous range of mathematical objects. The examples below serve both as illustrations of the definition and as a supply of concrete groups that will reappear throughout the theory — as test cases for theorems, sources of counterexamples, and motivation for new definitions.

[example:The Integers Under Addition] The most familiar group is $(\mathbb{Z}, +)$, the integers under addition. Associativity and the existence of an identity (zero) and inverses (negatives) are standard properties of integer arithmetic. Addition is commutative, so $\mathbb{Z}$ is abelian. The natural numbers $(\mathbb{N}, +)$ do *not* form a group: while addition is associative and $0$ is an identity (taking $\mathbb{N} = \{0, 1, 2, \ldots\}$), the element $1$ has no additive inverse in $\mathbb{N}$. This failure illustrates that the inverse axiom is genuinely restrictive — it forces the algebraic system to be "large enough" to undo every operation. [/example]

[example:Cyclic Groups And Modular Arithmetic] Fix $n > 1$. The set $\mathbb{Z}/n\mathbb{Z} = \{0, 1, 2, \ldots, n-1\}$ with addition modulo $n$ forms a finite abelian group of order $n$. Every element $a$ has a unique inverse $-a \pmod{n}$. This group is *cyclic*: every element is a power of the generator $1$. The multiplicative group of a [finite field](/page/Field) $\mathbb{F}_p$ (where $p$ is prime) consists of the nonzero residue classes $\{1, 2, \ldots, p-1\}$ under multiplication modulo $p$. This is an abelian group of order $p - 1$, and it is also cyclic — a non-trivial fact whose proof requires the theory of [polynomials](/page/Polynomial) over finite fields. [/example]