# 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] [example:Symmetric And Dihedral Groups] The set of all bijections $\sigma : S \to S$ from a finite set $S$ to itself forms a group under composition, called the *symmetric group* $S_n$ (where $|S| = n$). The identity is the identity [function](/page/Function), and every bijection has an inverse function. For $n \geq 3$, $S_n$ is non-abelian: for instance, in $S_3$, composing $(1\;2)$ then $(1\;3)$ gives $(1\;3\;2)$, while composing in the reverse order gives $(1\;2\;3)$. The set of all symmetries of a regular $n$-gon (rotations and reflections) forms the *dihedral group* $D_n$, of order $2n$. For $n = 3$, this recovers the six symmetries of the equilateral triangle from the opening motivation. The dihedral groups are among the simplest non-abelian groups and serve as a testing ground for many concepts: subgroups, normality, and conjugacy all exhibit interesting behaviour in $D_n$. [/example] [example:Matrix Groups] The set of all invertible $n \times n$ matrices with real entries forms the *general linear group* $\operatorname{GL}_n(\mathbb{R})$ under matrix multiplication. The identity is the identity matrix $I$, and every invertible matrix has an inverse by definition. The group is non-abelian for $n \geq 2$. Important subgroups include the *orthogonal group* $O(n)$ (matrices satisfying $M^T M = I$), the *special orthogonal group* $SO(n)$ (orthogonal matrices with $\det M = +1$), and the *special linear group* $\operatorname{SL}_n(\mathbb{R})$ (matrices with $\det M = 1$). These matrix groups connect group theory to [geometry](/page/Geometry) and [differential equations](/page/Ordinary%20Differential%20Equation): the group $SO(3)$ describes rotations in three-dimensional space, and [Lie groups](/page/Lie%20Group) — smooth manifolds that are also groups — are the natural setting for continuous symmetry. [/example] [example:The Trivial Group] The one-element set $\{e\}$ with the operation $e \cdot e = e$ is a group — the *trivial group*. It is the unique group of order $1$, and it is a subgroup of every group. The trivial group plays the same role in group theory that $\{0\}$ plays in [linear algebra](/page/Linear%20Algebra): it is the degenerate case that must be checked but rarely exhibits interesting behaviour. [/example] ## Subgroups and the Subgroup Criterion Many of the deepest questions in group theory concern the internal structure of a group: which subsets inherit the group operation? How do these subsets sit inside the ambient group? The concept of a *subgroup* makes these questions precise. The motivating observation is that certain subsets of a group "close up" under the group operation — applying the operation to elements of the subset always stays within the subset. The even integers $2\mathbb{Z}$ inside $\mathbb{Z}$, the rotations inside the symmetries of a polygon, and the matrices with determinant $1$ inside $\operatorname{GL}_n(\mathbb{R})$ all exhibit this property. Identifying exactly which conditions a subset must satisfy to form a group in its own right leads to the subgroup definition. [definition:Subgroup] Let $G$ be a group. A subset $H \subseteq G$ is a **subgroup** of $G$, written $H \leq G$, if: 1. $H$ is non-empty. 2. $H$ is closed under the group operation: if $a, b \in H$, then $ab \in H$. 3. $H$ contains inverses: if $a \in H$, then $a^{-1} \in H$. [/definition] If $H \leq G$, then $H$ is itself a group under the operation of $G$: associativity holds because it holds in $G$, the identity of $G$ lies in $H$ (pick any $a \in H$; then $a^{-1} \in H$ by condition 3, and $e = a a^{-1} \in H$ by condition 2), and every element of $H$ has an inverse in $H$ by condition 3. In practice, verifying all three conditions can be streamlined. The following criterion combines closure under the operation and closure under inversion into a single check, which is especially convenient when working with concrete groups. [quotetheorem:932] The criterion is a workhorse: it reduces three checks (non-emptiness, closure, inverses) to non-emptiness plus a single algebraic condition. However, it is not always the most efficient test. For finite subsets, an even simpler criterion holds: a non-empty finite subset $H \subseteq G$ is a subgroup if and only if $H$ is closed under the group operation (inverses come for free from finiteness, since repeated multiplication of any element must eventually cycle back to the identity). [example:Subgroups Of The Integers] The subgroups of $(\mathbb{Z}, +)$ are exactly the sets $n\mathbb{Z} = \{nk : k \in \mathbb{Z}\}$ for $n \geq 0$. To verify that $n\mathbb{Z} \leq \mathbb{Z}$, apply the subgroup criterion: for $na, nb \in n\mathbb{Z}$, we have $na - nb = n(a - b) \in n\mathbb{Z}$. Conversely, any subgroup of $\mathbb{Z}$ is of this form — a fact that follows from the [division algorithm](/page/Division%20Algorithm) and underpins the theory of [cyclic groups](/page/Cyclic%20Group). These subgroups form a containment chain: $\mathbb{Z} \supseteq 2\mathbb{Z} \supseteq 4\mathbb{Z} \supseteq \ldots$ [/example] [example:Subgroups Of S3] The symmetric group $S_3$ (permutations of three elements) has exactly six subgroups: - The trivial subgroup $\{e\}$ (order $1$). - Three subgroups of order $2$: $\{e, (1\;2)\}$, $\{e, (1\;3)\}$, and $\{e, (2\;3)\}$, each generated by a single transposition. - The alternating group $A_3 = \{e, (1\;2\;3), (1\;3\;2)\}$ (order $3$), consisting of the even permutations. - The full group $S_3$ itself (order $6$). That there are no subgroups of order $4$ or $5$ is a consequence of [Lagrange's theorem](/theorems/841), which we develop next. [/example] ## Cosets and Lagrange's Theorem The first deep structural constraint in group theory is the observation that the order of a subgroup must divide the order of the group. This is not obvious from the definition — there is no a priori reason why a subset that happens to be closed under the operation should have a size that divides $|G|$. The key idea is to partition the group into equally-sized pieces called *cosets*. To state the result precisely, we need to name the quantity being divided. [definition:Order Of A Group] The **order** of a group $G$, denoted $|G|$, is the cardinality of the underlying set. The **order** of an element $g \in G$ is the smallest positive integer $n$ such that $g^n = e$, or $\infty$ if no such $n$ exists. The order of $g$ is denoted $\operatorname{ord}(g)$ or $|g|$. [/definition] The mechanism behind Lagrange's theorem is the *coset*: a uniform translation of a subgroup across the group. Cosets arise naturally when one attempts to describe the "position" of an element relative to a subgroup — two elements $g_1$ and $g_2$ are in the same position relative to $H$ precisely when $g_1^{-1} g_2 \in H$, which is the condition for them to lie in the same coset. [definition:Coset] Let $G$ be a group and $H \leq G$. A **left coset** of $H$ in $G$ is a set of the form $gH = \{gh : h \in H\}$ for some $g \in G$. A **right coset** is $Hg = \{hg : h \in H\}$. The number of distinct left cosets of $H$ in $G$ is the **index** of $H$ in $G$, written $|G : H|$. [/definition] Two left cosets $g_1 H$ and $g_2 H$ are either disjoint or equal — they are equal precisely when $g_1^{-1} g_2 \in H$. This means the left cosets partition $G$ into non-overlapping pieces. Moreover, the [function](/page/Function) $h \mapsto gh$ is a bijection from $H$ to $gH$, so every coset has exactly $|H|$ elements. Since $G$ is the disjoint union of its cosets, each of size $|H|$, the order of $G$ must be a multiple of $|H|$. [quotetheorem:841] Lagrange's theorem is the single most frequently invoked result in elementary group theory. Its power lies in the *constraints* it imposes: if $|G| = 12$, then the only possible subgroup orders are $1, 2, 3, 4, 6, 12$. A subgroup of order $5$ or $7$ cannot exist. This immediately restricts the structure of $G$ and is the starting point for most classification arguments. However, the converse of Lagrange's theorem is *false*: a group of order $12$ need not have a subgroup of every divisor order. The alternating group $A_4$ (order $12$) has no subgroup of order $6$, demonstrating that divisibility of orders is necessary but not sufficient for the existence of subgroups. An immediate corollary connects element orders to the group order, which is the bridge between the "internal" notion of how an element behaves under repeated multiplication and the "external" constraint of the group's size. [quotetheorem:783] This result has an elegant consequence: every group of prime order is cyclic. If $|G| = p$ is prime, then every non-identity element $g$ has order dividing $p$, so $\operatorname{ord}(g) = p$ (since $\operatorname{ord}(g) > 1$), meaning $g$ generates all of $G$. This is recorded as [Prime Order Implies Cyclic](/theorems/784) in the theorem database. The result also underpins Fermat's little theorem in number theory: for prime $p$ and $a \not\equiv 0 \pmod{p}$, the element $a$ in the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^\times$ (which has order $p - 1$) satisfies $a^{p-1} \equiv 1 \pmod{p}$. ### The failure of the converse of Lagrange's theorem It is natural to ask whether the converse holds: if $d$ divides $|G|$, must $G$ have a subgroup of order $d$? For abelian groups the answer is yes (a consequence of the [fundamental theorem of finite abelian groups](/page/Abelian%20Group)), and [Cauchy's theorem](/theorems/797) guarantees a subgroup of order $p$ for every prime $p$ dividing $|G|$. But the full converse fails. [example:No Subgroup Of Order Six In A4] The alternating group $A_4$ consists of the twelve even permutations of $\{1, 2, 3, 4\}$: the identity, eight $3$-cycles, and three products of disjoint transpositions. Although $6$ divides $|A_4| = 12$, there is no subgroup of order $6$. To see why, suppose $H \leq A_4$ with $|H| = 6$. Then $|A_4 : H| = 2$, so $H$ is normal in $A_4$ (every subgroup of index $2$ is normal). But $H$ would contain all elements of $A_4$ whose square is the identity — namely the three products of disjoint transpositions — plus the identity, giving at least four elements, and to reach order $6$ it would need two of the eight $3$-cycles. However, any $3$-cycle $\sigma$ satisfies $\sigma^2 = \sigma^{-1}$, which is also a $3$-cycle; so if $\sigma \in H$ then $\sigma^{-1} \in H$, meaning $3$-cycles enter in pairs. This gives $|H| \geq 4 + 2 = 6$, which works, but checking all possible pairs of $3$-cycles reveals that no such $H$ is closed under composition. This concrete failure motivates the Sylow theorems, which provide a partial converse for prime-power divisors. [/example] ## Homomorphisms and Kernels The preceding sections studied the internal structure of a single group. The next question is: how do different groups relate to one another? Two groups may look superficially different — one may consist of numbers, the other of matrices or permutations — yet share the same algebraic structure. Making this precise requires a notion of *structure-preserving map*. The need for such maps arises concretely: the [function](/page/Function) $\phi : \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ sending each integer to its residue class preserves addition ($\phi(a + b) = \phi(a) + \phi(b)$), and this preservation is what makes modular arithmetic "compatible" with ordinary arithmetic. The determinant map $\det : \operatorname{GL}_n(\mathbb{R}) \to \mathbb{R}^\times$ satisfies $\det(AB) = \det(A)\det(B)$, converting matrix multiplication into scalar multiplication. Both are instances of the same concept. [definition:Group Homomorphism] Let $(G, \cdot_G)$ and $(K, \cdot_K)$ be groups. A **group homomorphism** is a function \begin{align*} \phi : G &\to K \\ g &\mapsto \phi(g) \end{align*} satisfying $\phi(a \cdot_G b) = \phi(a) \cdot_K \phi(b)$ for all $a, b \in G$. [/definition] Every homomorphism automatically preserves the identity ($\phi(e_G) = e_K$) and inverses ($\phi(a^{-1}) = \phi(a)^{-1}$). These are consequences of the defining property, not additional assumptions. When a homomorphism is also a bijection, it establishes that two groups are structurally indistinguishable — they have the same "shape" even if their elements look different. This is the strongest notion of sameness in group theory. [definition:Isomorphism] A group homomorphism $\phi : G \to K$ is an **isomorphism** if $\phi$ is a bijection. Two groups are **isomorphic**, written $G \cong K$, if there exists an isomorphism between them. [/definition] Isomorphic groups share all group-theoretic properties: they have the same order, the same number of elements of each order, the same subgroup lattice, and so on. [Cayley's theorem](/theorems/846) shows that every group is isomorphic to a subgroup of a symmetric group, demonstrating that the abstract group axioms do not produce anything "beyond" permutation groups — they merely provide a cleaner language for studying them. The most important invariant of a homomorphism is the set of elements it sends to the identity. This set measures how far the homomorphism is from being injective: a homomorphism is injective if and only if its kernel is trivial. More fundamentally, the kernel determines the "fibres" of the map — the sets $\phi^{-1}(\{k\})$ for $k \in \operatorname{im}(\phi)$ — and these fibres are precisely the cosets of the kernel. [definition:Kernel] The **kernel** of a group homomorphism $\phi : G \to K$ is the preimage of the identity: \begin{align*} \ker(\phi) = \{g \in G : \phi(g) = e_K\}. \end{align*} [/definition] The kernel is always a subgroup of $G$, but it satisfies a stronger condition: it is invariant under conjugation by every element of $G$. Subgroups with this property — normal subgroups — are exactly the subgroups that can serve as kernels, and they are exactly the subgroups for which the coset space $G/H$ inherits a well-defined group operation. ## Normal Subgroups and Quotient Groups The quotient construction is the process of "collapsing" a subgroup to the identity and obtaining a new, smaller group. For this to work, the coset multiplication $(g_1 H)(g_2 H) = (g_1 g_2)H$ must be well-defined — it must not depend on which representatives $g_1, g_2$ are chosen from their respective cosets. This well-definedness fails for arbitrary subgroups but holds precisely when left and right cosets coincide, which is the condition of *normality*. To see why an arbitrary subgroup does not suffice, consider $G = S_3$ and $H = \{e, (1\;2)\}$. The left coset $(1\;3)H = \{(1\;3), (1\;2\;3)\}$ and the right coset $H(1\;3) = \{(1\;3), (1\;3\;2)\}$ are different sets. Any attempt to define multiplication on cosets of $H$ leads to inconsistencies, since the same coset can be represented by different elements that give different products. Normality eliminates this ambiguity. [definition:Normal Subgroup] A subgroup $N$ of $G$ is **normal**, written $N \trianglelefteq G$, if $gNg^{-1} = N$ for all $g \in G$. Equivalently, $N$ is normal if and only if left and right cosets coincide: $gN = Ng$ for all $g \in G$. [/definition] Normality is a strictly weaker condition than being central (contained in the centre of $G$), and a strictly stronger condition than being a subgroup. In abelian groups, every subgroup is normal. In non-abelian groups, normality is a non-trivial condition: the subgroup $\{e, (1\;2)\}$ of $S_3$ is not normal, while the subgroup $A_3 = \{e, (1\;2\;3), (1\;3\;2)\}$ is. The fundamental fact linking normality to homomorphisms is that kernels are always normal, and conversely, every normal subgroup is the kernel of some homomorphism (namely, the quotient map $g \mapsto gN$). [quotetheorem:788] The converse direction — that every normal subgroup arises as a kernel — is established by the quotient construction itself: given $N \trianglelefteq G$, the set of cosets $G/N = \{gN : g \in G\}$ forms a group under the operation $(g_1 N)(g_2 N) = (g_1 g_2)N$, and the projection $\pi : G \to G/N$ defined by $\pi(g) = gN$ is a surjective homomorphism with $\ker(\pi) = N$. ## The Isomorphism Theorems The isomorphism theorems are the central structural results of group theory. They describe how quotients, subgroups, and homomorphisms interact, and they provide the main tools for decomposing groups into simpler pieces. Each theorem answers a specific structural question, and together they form a coherent picture of how the subgroup lattice of a group relates to the subgroup lattice of its quotients. The first isomorphism theorem answers the most basic question: what does the image of a homomorphism look like? The answer is that $\operatorname{im}(\phi)$ is always isomorphic to the quotient $G / \ker(\phi)$. This is the engine behind most concrete isomorphism constructions: to show that $G/N \cong H$, find a surjective homomorphism from $G$ to $H$ whose kernel is $N$. [quotetheorem:842] The first isomorphism theorem converts the problem of understanding homomorphic images into the problem of understanding quotients, which are often more tractable because they are described in terms of the internal structure of $G$ alone. Every time we compute a quotient group in practice — for instance, showing that $\mathbb{Z}/n\mathbb{Z}$ is the image of the reduction map $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ — we are implicitly using this theorem. The theorem also provides a bijection between cosets of the kernel and elements of the image, which is why "collapsing the kernel" and "looking at the image" are two descriptions of the same operation. The second isomorphism theorem addresses a more specific situation: when a subgroup $H$ and a normal subgroup $K$ coexist in $G$, how do $H$, $K$, and their interaction relate? [quotetheorem:843] This theorem is sometimes called the "diamond isomorphism theorem" because of the shape of the subgroup lattice diagram it describes: $G$ at the top, $HK$ and its subgroups forming a diamond below. The key insight is that $H$ and $K$ interact through their intersection $H \cap K$, and the theorem identifies this intersection as the precise obstruction to $H$ mapping injectively into $G/K$. In practice, the second isomorphism theorem is used to compute quotients of subgroups without having to construct the quotient directly. The correspondence theorem (sometimes called the fourth isomorphism theorem or lattice isomorphism theorem) provides a global perspective: it describes the relationship between *all* subgroups of $G$ that contain $N$ and *all* subgroups of $G/N$. [quotetheorem:854] The correspondence theorem means that passing to a quotient $G/N$ does not destroy information about the subgroups "above" $N$ — it merely compresses them. Every subgroup of $G/N$ lifts to a unique subgroup of $G$ containing $N$, and this lifting preserves inclusion, normality, and index. This is the key tool for inductive arguments in group theory: to study $G$, factor out a normal subgroup $N$, study the (smaller) quotient $G/N$ using the correspondence theorem, and then "lift" the results back to $G$. ## Group Actions The preceding theory developed the internal algebra of groups: subgroups, quotients, and homomorphisms. Group actions connect this abstract machinery to the rest of mathematics by formalising the idea that a group "moves" or "transforms" the elements of a set. Rotations act on points of a plane, permutations act on the elements being permuted, and matrices act on vectors. The power of the formalism is that the same group-theoretic tools — cosets, stabilisers, orbit decompositions — apply uniformly to all of these situations. The need for a formal definition becomes clear when one tries to count. How many distinct necklaces can be made from $n$ beads of $k$ colours? The answer depends on which rearrangements are considered "the same" — that is, on which group of symmetries acts on the set of colourings. Without the language of group actions, each such counting problem requires its own ad hoc argument; with it, a single framework ([Burnside's lemma](/page/Burnside%27s%20Lemma)) handles them all. [definition:Group Action] Let $G$ be a group and $X$ a set. A **left group action** of $G$ on $X$ is a function \begin{align*} G \times X &\to X \\ (g, x) &\mapsto g \cdot x \end{align*} satisfying: 1. $e \cdot x = x$ for all $x \in X$. 2. $(gh) \cdot x = g \cdot (h \cdot x)$ for all $g, h \in G$ and $x \in X$. [/definition] Condition 2 ensures that the action is "compatible" with the group operation: acting by $gh$ is the same as first acting by $h$, then by $g$. This compatibility means that a group action is equivalent to a homomorphism $\rho : G \to \operatorname{Sym}(X)$ from $G$ to the symmetric group of $X$, where $\rho(g)$ is the bijection $x \mapsto g \cdot x$. This perspective — viewing an action as a *representation* of $G$ by permutations — connects group actions to [representation theory](/page/Representation%20Theory). [example:Standard Group Actions] The symmetric group $S_n$ acts on $\{1, 2, \ldots, n\}$ by $\sigma \cdot i = \sigma(i)$. This is the *defining action* of $S_n$. The dihedral group $D_n$ acts on the vertices of a regular $n$-gon by rotations and reflections. The general linear group $\operatorname{GL}_n(\mathbb{R})$ acts on $\mathbb{R}^n$ by matrix-vector multiplication: $A \cdot v = Av$. Every group $G$ acts on itself by *left multiplication*: $g \cdot x = gx$ for $g, x \in G$. This action is always faithful (only the identity fixes every element), and it is the action used in the proof of [Cayley's theorem](/theorems/846). Every group $G$ also acts on itself by *conjugation*: $g \cdot x = gxg^{-1}$. The orbits of this action are the *conjugacy classes* of $G$, and the stabiliser of $x$ is the *centraliser* $C_G(x)$. This action is the key to the [class equation](/page/Class%20Equation) and many counting arguments. [/example] ## Orbits, Stabilisers, and Counting A group action partitions the set $X$ into orbits — the maximal subsets within which elements can be moved to one another by the action. Understanding the size of these orbits is a fundamental problem, and the orbit-stabiliser theorem provides the answer by relating orbit sizes to the sizes of certain subgroups. The orbit of a point records *where* it can be moved; the stabiliser records *what* fixes it. These two pieces of information are complementary: a larger stabiliser means the element has more symmetries, which means fewer elements in its orbit. The orbit-stabiliser theorem makes this tradeoff precise. [definition:Orbit] Let $G$ act on a set $X$. The **orbit** of $x \in X$ under $G$ is \begin{align*} \operatorname{Orb}(x) = \{g \cdot x : g \in G\}. \end{align*} [/definition] [definition:Stabiliser] The **stabiliser** of $x \in X$ is the subgroup \begin{align*} \operatorname{Stab}(x) = \{g \in G : g \cdot x = x\}. \end{align*} That $\operatorname{Stab}(x)$ is indeed a subgroup follows from the action axioms: if $g, h$ both fix $x$, then $gh$ fixes $x$ (by condition 2), and $g^{-1}$ fixes $x$ (apply $g^{-1}$ to both sides of $g \cdot x = x$). [/definition] [quotetheorem:845] The orbit-stabiliser theorem is the group-action analogue of [Lagrange's theorem](/theorems/841): Lagrange says $|G| = |H| \cdot |G : H|$ for a subgroup $H$, and the orbit-stabiliser theorem says $|G| = |\operatorname{Stab}(x)| \cdot |\operatorname{Orb}(x)|$, which is Lagrange applied to the subgroup $\operatorname{Stab}(x)$. The deep content is that the *index* $|G : \operatorname{Stab}(x)|$ equals the *orbit size* — that is, the cosets of $\operatorname{Stab}(x)$ are in natural bijection with the points of the orbit. This bijection is the map $g\operatorname{Stab}(x) \mapsto g \cdot x$. An important application is the *class equation*. When $G$ acts on itself by conjugation, the orbits are the conjugacy classes $C_1, \ldots, C_k$, and the orbit-stabiliser theorem gives $|C_i| = |G : C_G(x_i)|$ for any representative $x_i \in C_i$. Summing over conjugacy classes: \begin{align*} |G| = |Z(G)| + \sum_{i} |G : C_G(x_i)|, \end{align*} where the sum runs over conjugacy classes of size greater than $1$ and $Z(G)$ is the centre of $G$. The class equation is the key to proving that the centre of a $p$-group is non-trivial ([Centre of a p-Group Is Nontrivial](/theorems/799)), which in turn is the starting point for the Sylow theorems. [example:Orbit Stabiliser In Action] Let $G = D_4$ (symmetries of a square, order $8$) act on the four vertices $\{1, 2, 3, 4\}$ of the square. The orbit of vertex $1$ is all four vertices (any vertex can be sent to any other by a suitable symmetry), so $|\operatorname{Orb}(1)| = 4$. By the orbit-stabiliser theorem, $|\operatorname{Stab}(1)| = 8/4 = 2$. Indeed, $\operatorname{Stab}(1)$ consists of the identity and the reflection through the diagonal passing through vertex $1$. [/example] ## Structure and Classification The ultimate goal of group theory is to understand the structure of all groups — ideally, to classify them. For finite abelian groups, this goal is completely achieved. For non-abelian groups, the situation is vastly more complex, but powerful partial results constrain what is possible. The fundamental theorem of finite abelian groups provides a unique decomposition: every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order. This means the structure of a finite abelian group is entirely determined by a list of integers (the prime power orders of the cyclic factors), and two finite abelian groups are isomorphic if and only if they have the same list. No analogous classification exists for non-abelian groups. For non-abelian groups, the deepest structural constraints come from the Sylow theorems, which provide a partial converse to [Lagrange's theorem](/theorems/841) for prime-power divisors. If $p^k$ divides $|G|$, then $G$ has a subgroup of order $p^k$ (a *Sylow $p$-subgroup* when $p^k$ is the largest power of $p$ dividing $|G|$). Moreover, the number of Sylow $p$-subgroups is congruent to $1$ modulo $p$ and divides $|G|$. These constraints are often strong enough to determine the structure of groups of small order and to prove that certain groups must have normal subgroups — which is the starting point for the [Jordan-Hölder theorem](/page/Jordan-H%C3%B6lder%20Theorem) and the classification of finite simple groups. Additional key results in the database include [Cauchy's theorem](/theorems/797) (every finite group whose order is divisible by a prime $p$ contains an element of order $p$), [Cayley's theorem](/theorems/846) (every group embeds into a symmetric group), the [classification of groups of order $p^2$](/theorems/818) (all abelian, either cyclic or a direct product of two copies of $\mathbb{Z}/p\mathbb{Z}$), and the [simplicity of $A_5$](/theorems/819) (the smallest non-abelian simple group, which is the obstruction to solving the quintic by radicals). ## Common Techniques Several recurring methods appear throughout group theory. Recognising them explicitly helps both in solving problems and in reading proofs. **Lagrange counting.** Given a finite group $G$, use [Lagrange's theorem](/theorems/841) to restrict which subgroup orders, element orders, or index values are possible. This is the first tool to deploy in any structural argument about finite groups. **The quotient-and-lift strategy.** To study a group $G$, find a normal subgroup $N$, study the quotient $G/N$ (which is smaller), and use the [correspondence theorem](/theorems/854) to lift information back to $G$. This reduces problems about $G$ to problems about two simpler objects: $N$ and $G/N$. **Constructing homomorphisms to identify quotients.** To show that $G/N \cong H$, find a surjective homomorphism $\phi : G \to H$ with $\ker(\phi) = N$ and invoke the [first isomorphism theorem](/theorems/842). This is almost always more efficient than constructing an isomorphism $G/N \to H$ directly. **Orbit counting.** To count objects "up to symmetry," define a group action and apply the orbit-stabiliser theorem or Burnside's lemma. The key step is identifying the right group and the right set for the action. **Conjugation arguments.** Many structural results (the class equation, normality tests, Sylow theory) use the conjugation action $g \cdot x = gxg^{-1}$. The orbits are conjugacy classes, and the stabiliser of $x$ is the centraliser $C_G(x)$. ## References - Dummit, D. S. and Foote, R. M., *Abstract Algebra*, 3rd edition, Wiley (2004). - Gallian, J. A., *Contemporary Abstract Algebra*, 10th edition, Cengage (2021). - Humphreys, J. F., *A Course in Group Theory*, Oxford University Press (1996). - Lang, S., *Algebra*, 3rd edition, Springer (2002). - Rotman, J. J., *An Introduction to the Theory of Groups*, 4th edition, Springer (1995). - Serre, J.-P., *Finite Groups: An Introduction*, International Press (2016).