Finite group
In mathematics, particularly in the field of abstract algebra, a finite group is a group that consists of a finite number of elements under a binary operation satisfying closure, associativity, the existence of an identity element, and inverses for each element.[1] The number of elements in such a group G, denoted |G|, is called the order of the group.[2] Finite groups are essential structures for modeling symmetries in mathematical objects, physical systems, and combinatorial problems, often arising in the study of permutations, rotations, and modular arithmetic.[3] Prominent examples include the cyclic group \mathbb{Z}/p\mathbb{Z} of order p (where p is prime), which is generated by a single element under addition modulo p; the symmetric group S_n, which comprises all permutations of n objects and has order n!; and the alternating group A_n, consisting of even permutations with order n!/2 for n \geq 2.[2] A foundational result, Lagrange's theorem, asserts that if H is a subgroup of a finite group G, then the order of H divides the order of G, implying that the order of any element in G also divides |G|.[2] The theory of finite groups encompasses deep results on their structure and classification, with simple groups serving as the "atoms" or indecomposable building blocks, unable to be expressed as nontrivial quotients of smaller groups.[4] The classification of finite simple groups (CFSG), one of the most significant theorems in modern mathematics, proves that every finite simple group belongs to one of 18 infinite families (such as cyclic groups of prime order, alternating groups A_n for n \geq 5, and groups of Lie type) or one of 26 exceptional "sporadic" groups, like the Monster group of order approximately $8 \times 10^{53}.[4] This classification, developed over decades by more than 100 mathematicians and spanning over 15,000 pages of proofs, has profound implications for group theory, representation theory, and applications in physics, such as particle symmetries.[4]Fundamentals
Definition
In abstract algebra, a group is a nonempty set G equipped with a binary operation \cdot: G \times G \to G that satisfies four axioms: closure (for all a, b \in G, a \cdot b \in G); associativity (for all a, b, c \in G, (a \cdot b) \cdot c = a \cdot (b \cdot c)); identity (there exists an element e \in G such that for all a \in G, a \cdot e = e \cdot a = a); and invertibility (for each a \in G, there exists a^{-1} \in G such that a \cdot a^{-1} = a^{-1} \cdot a = e)./02%3A_Groups_I/2.02%3A_Definition_of_a_Group)[5][6] A finite group is a group whose underlying set G has finite cardinality, denoted |G| = n where n is a positive integer called the order of the group.[7][8] Groups may use multiplicative notation (with operation \cdot and identity e) or additive notation (with operation + and identity $0), depending on context; for example, the set \mathbb{Z}/n\mathbb{Z} of integers modulo n under addition forms a finite group of order n./02%3A_Groups_I/2.02%3A_Definition_of_a_Group)[7] The trivial group consists of a single element \{e\} satisfying all group axioms, with order |G| = 1.[6][8] Finite groups contrast with infinite groups, where |G| is infinite, though both share the same axiomatic structure.[7]Order of elements and groups
In a group G with identity element e, the order of an element g \in G, denoted o(g) or |g|, is the smallest positive integer k such that g^k = e, provided such a k exists; otherwise, the order is defined to be infinite.[9] In the context of finite groups, every element has finite order, as the powers of g cannot cycle indefinitely within a set of bounded size.[9] Moreover, in any finite group G of order |G|, the order of every element divides |G|, a property that highlights the structural constraints imposed by finiteness.[10] The cyclic subgroup generated by an element g \in G, denoted \langle g \rangle, consists of all integer powers of g: \langle g \rangle = \{ g^k \mid k \in \mathbb{Z} \}.[9] If g has finite order k, then \langle g \rangle = \{ e, g, g^2, \dots, g^{k-1} \}, and the order of this subgroup equals k, the order of g.[9] This subgroup provides insight into the local structure around g, as its size directly reflects how many distinct powers g produces before returning to the identity. The orders of elements thus reveal much about the overall group structure, with elements of larger orders generating larger cyclic subgroups that embed within G. For example, consider the additive cyclic group \mathbb{Z}_n of integers modulo n, which has order n. The order of an element m \in \mathbb{Z}_n (with $0 \leq m < n) is n / \gcd(m, n), the smallest positive integer k such that k m \equiv 0 \pmod{n}.[11] Thus, generators like m = 1 have order n, while elements sharing factors with n yield smaller orders; for instance, in \mathbb{Z}_{12}, the order of 4 is 3 since $3 \cdot 4 = 12 \equiv 0 \pmod{12}. In the symmetric group S_3 of order 6, which permutes three elements, the identity has order 1, transpositions like (1\ 2) have order 2 (as (1\ 2)^2 = e), and 3-cycles like (1\ 2\ 3) have order 3 (as (1\ 2\ 3)^3 = e).[12] These orders—1, 2, and 3—all divide 6, illustrating the general relation in finite groups.Basic theorems
Lagrange's theorem
Lagrange's theorem asserts that if G is a finite group and H is a subgroup of G, then the order of H, denoted |H|, divides the order of G, denoted |G|.[13] The proof relies on the notion of cosets. For a subgroup H of G, a left coset of H is a set of the form gH = \{ gh \mid h \in H \} where g \in G. A right coset is defined analogously as Hg = \{ hg \mid h \in H \}.[13] Consider the relation \sim on G given by g_1 \sim g_2 if and only if g_1 H = g_2 H (equivalently, g_1^{-1} g_2 \in H). This relation is reflexive, symmetric, and transitive, hence an equivalence relation. The equivalence classes are the distinct left cosets of H, which partition G into disjoint sets. Moreover, any two distinct left cosets are disjoint, and every element of G belongs to exactly one left coset. Each left coset has cardinality |H|, as there is a bijection between H and gH given by left multiplication by g.[13] Let [G : H] denote the number of distinct left cosets of H in G, called the index of H in G. Then G is the disjoint union of these [G : H] cosets, so |G| = [G : H] \cdot |H|. Since [G : H] is a positive integer, it follows that |H| divides |G|.[13] A direct consequence is that the order of any element g \in G divides |G|. Indeed, the cyclic subgroup \langle g \rangle generated by g has order equal to the order of g, and thus divides |G| by the theorem.[13]Consequences of Lagrange's theorem
Lagrange's theorem implies that the order of every subgroup divides the order of the group, but the converse does not hold in general. For instance, the alternating group A_4 has order 12, yet it contains no subgroup of order 6.[14] A significant structural consequence arises for subgroups of small index. Specifically, any subgroup H of a finite group G with index [G : H] = 2 is normal in G. This follows because the left and right cosets of H coincide, as there are only two cosets: H itself and its complement G \setminus H.[15] Another key implication is Cauchy's theorem, which states that if p is a prime dividing the order of a finite group G, then G contains an element of order p.[16] In the context of number theory, Lagrange's theorem applies to the multiplicative group U(n) of integers modulo n that are coprime to n, which has order \phi(n), where \phi is Euler's totient function. For any g \in U(n), the order of g divides \phi(n), so g^{\phi(n)} \equiv 1 \pmod{n}. This is known as Euler's theorem.[17] When n = p is prime, \phi(p) = p-1, yielding Fermat's little theorem: if p does not divide g, then g^{p-1} \equiv 1 \pmod{p}. This is a special case of Euler's theorem.[18]Examples
Permutation groups
A permutation group is a subgroup of the symmetric group on a finite set, where the group operation is composition of permutations.[19] The symmetric group S_n, or \mathrm{Sym}(n), consists of all bijections from a set of n elements to itself and has order n!.[20] The alternating group A_n is the subgroup of S_n generated by even permutations, which are those expressible as a product of an even number of transpositions (2-cycles).[21] It has index 2 in S_n, making it a normal subgroup of order n!/2.[22] For n \geq 5, A_n is simple, meaning it has no nontrivial normal subgroups.[23] A transposition is a 2-cycle that swaps two elements while fixing the rest, and every permutation in S_n can be uniquely decomposed (up to ordering) into a product of disjoint cycles, including fixed points as 1-cycles.[24] For example, the symmetric group S_3 has order 6 and is non-abelian, as compositions like (1\ 2)(2\ 3) = (1\ 2\ 3) and (2\ 3)(1\ 2) = (1\ 3\ 2) do not commute.[25] Another example is the dihedral group D_n, which is a permutation group of order $2n realizing the rotational and reflection symmetries of a regular n-gon.[26] Permutation groups also model puzzles like the Rubik's Cube, invented in 1974, where the set of legal moves generates a subgroup of the direct product of symmetric groups on edges and corners, with order approximately $4.3 \times 10^{19}.[27]Cyclic groups
A cyclic group is a group that can be generated by a single element, known as a generator. For a finite cyclic group G of order n, there exists an element g \in G such that G = \langle g \rangle = \{e, g, g^2, \dots, g^{n-1}\}, where g^n = e is the identity element.) All finite cyclic groups of order n are isomorphic to one another and can be represented additively as the cyclic group \mathbb{Z}/n\mathbb{Z} of integers modulo n under addition. Multiplicatively, they are isomorphic to the group of nth roots of unity in the complex numbers, consisting of solutions to z^n = 1.[28] Every subgroup of a finite cyclic group of order n is itself cyclic, and for each positive divisor d of n, there exists exactly one subgroup of order d, generated by g^{n/d}.[29] A finite group G of order n is cyclic if and only if, for every divisor d of n, G contains exactly \phi(d) elements of order d, where \phi denotes Euler's totient function.[29] Representative examples include the additive group \mathbb{Z}/12\mathbb{Z}, which models clock arithmetic where addition is performed modulo 12 and generated by 1. Another is the group of rotations of a regular n-gon about its center, which is cyclic of order n under composition.[28]Finite abelian groups
Finite abelian groups extend the structure of cyclic groups by allowing direct products of multiple cyclic components, enabling a complete classification up to isomorphism. Unlike cyclic groups, which are generated by a single element, finite abelian groups can have more complex decompositions while remaining commutative. The key result characterizing these groups is the Fundamental Theorem of Finite Abelian Groups, which states that every finite abelian group G is isomorphic to a direct product of cyclic groups of prime power order:G \cong \mathbb{Z}/p_1^{k_1}\mathbb{Z} \times \mathbb{Z}/p_1^{k_2}\mathbb{Z} \times \cdots \times \mathbb{Z}/p_m^{k_m}\mathbb{Z} \times \cdots,
where the primes p_i may repeat and the exponents k_j are positive integers.[30] This theorem provides a canonical form that uniquely determines the group's isomorphism class based on its order.[30] The primary decomposition aspect of the theorem decomposes G into its Sylow p-subgroups for each prime p dividing |G|, yielding
G \cong \bigoplus_p G_p,
where each G_p is the p-primary component, a finite abelian p-group isomorphic to a direct sum of cyclic groups of orders powers of p: G_p \cong \mathbb{Z}/p^{a_1}\mathbb{Z} \oplus \mathbb{Z}/p^{a_2}\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/p^{a_r}\mathbb{Z} with a_1 \geq a_2 \geq \cdots \geq a_r \geq 1.[30] These components are independent, as the order of G factors into distinct prime powers. The elementary divisors form refers to the collection of these prime power orders p^{a_i}, which fully specify the group up to isomorphism when sorted appropriately.[31] In contrast, the invariant factors decomposition expresses G as a direct product of cyclic groups \mathbb{Z}/m_1\mathbb{Z} \times \mathbb{Z}/m_2\mathbb{Z} \times \cdots \times \mathbb{Z}/m_s\mathbb{Z}, where m_1 divides m_2, m_2 divides m_3, and so on, up to m_s, and the product of the m_i equals |G|.[31] This form is unique and often more compact for computation, though deriving it from the elementary divisors involves combining prime powers across different primes while preserving the divisibility condition. Both decompositions are equivalent representations of the same theorem, with transformations between them possible via prime factorization.[31] A representative example is the Klein four-group V_4, which has order 4 and is isomorphic to \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}. This group is non-cyclic, as no single element generates it—all non-identity elements have order 2—and its primary decomposition consists of two copies of the cyclic group of order 2. In invariant factors form, it remains \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, since 2 divides 2. Another illustration is the abelian group of order 8 given by \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}, with elementary divisors 2 and 4, contrasting with the cyclic \mathbb{Z}/8\mathbb{Z}. Homomorphisms between finite abelian groups G and H are determined by the primary decompositions: \Hom(G, H) \cong \prod_p \Hom(G_p, H_p), where each component homomorphism respects the cyclic sum structure. Endomorphisms of G, i.e., \End(G), form a ring whose structure mirrors the direct sum of matrix rings over the integers modulo the relevant orders, facilitating computations of the group's module-like properties.[30]
p-groups
A finite p-group is a finite group G whose order |G| is a power of a prime p, that is, |G| = pk for some integer k ≥ 1.[32] Every nontrivial finite p-group has a nontrivial center Z(G), meaning |Z(G)| ≥ p.[33] Moreover, every maximal subgroup of a finite p-group is normal and has index p.[32] A key structural property is that every finite p-group of order pk possesses a normal subgroup of order pm for each m with 1 ≤ m ≤ k; in fact, the number of such subgroups depends on the group's structure, but their existence follows from iteratively applying the nontrivial center property to build a chain of normal subgroups.[33] Prominent examples include the quaternion group Q8 of order 8 (p=2), which is non-abelian and has presentation <a,b | a4=1, a2=b2, b-1a(b)=a-1>; its center is <a2> of order 2, and it has three subgroups of order 4, all normal and cyclic.[32] Another example is the elementary abelian p-group (Z/pZ)k, which is abelian, isomorphic to the vector space (Fp)k under addition, where every non-identity element has order p and all subgroups are normal.[32] The Frattini subgroup Φ(G) of a finite p-group G is the intersection of all its maximal subgroups, which coincides with the subgroup generated by all commutators [g,h] and all pk-th powers gpk for g ∈ G; notably, G/Φ(G) is elementary abelian of order pd, where d is the minimal number of generators of G.[32] Finite p-groups admit a chief series, a maximal normal series 1 = N0 ⊴ N1 ⊴ ⋯ ⊴ Nk = G where each factor N*i+1/Ni is a minimal normal subgroup of G/Ni and has order p, reflecting the p-group's nilpotent structure with chief factors that are elementary abelian of rank 1.[32] The Burnside basis theorem states that if G is a finite p-group, then the minimal number of generators d(G) equals the dimension of the elementary abelian group G/Φ(G) over Fp, and any generating set of G maps onto a basis of G/Φ(G) if and only if it generates G.[32]Groups of Lie type
Groups of Lie type are finite groups constructed as the fixed points under certain endomorphisms of algebraic groups defined over finite fields, serving as discrete analogues of classical Lie groups. These groups emerge from the rational points of a reductive algebraic group G defined over a finite field \mathbb{F}_q, where q is a power of a prime, typically taking the form G(\mathbb{F}_q) or subgroups thereof, such as the special linear group \mathrm{SL}(n, q) or its projective version \mathrm{PSL}(n, q). The foundational construction relies on the structure of semisimple Lie algebras over the complex numbers, transported to characteristic p via a Chevalley basis, which allows the generation of these finite groups uniformly across different types. The untwisted groups of Lie type, known as Chevalley groups, are classified by Dynkin diagrams corresponding to simple Lie algebras and include families such as A_n(q) = \mathrm{PSL}(n+1, q), the projective special linear groups; B_n(q), the odd-dimensional orthogonal groups over \mathbb{F}_q; C_n(q), the symplectic groups \mathrm{Sp}(2n, q); D_n(q), the even-dimensional orthogonal groups; and the exceptional types E_6(q), E_7(q), E_8(q), F_4(q), and G_2(q). Twisted variants, introduced to capture additional simple groups, arise by applying a Frobenius endomorphism (field automorphism combined with a graph automorphism) to the Chevalley group; prominent examples are the Suzuki groups ^2B_2(q) for q = 2^{2m+1}, the Ree groups of type ^2G_2(q) for q = 3^{2m+1}, and ^2F_4(q) for q = 2^{2m+1}. These twisted groups fill out the complete list of non-abelian finite simple groups of Lie type, excluding the alternating and sporadic families.[34] The orders of these groups follow explicit formulas derived from the Weyl group and root system structure. For instance, the order of \mathrm{PSL}(2, q) is \frac{q(q-1)(q+1)}{d}, where d = \gcd(2, q-1), reflecting the quotient of \mathrm{SL}(2, q) by its center; this yields q(q^2 - 1)/2 when q is odd and q(q^2 - 1) when q is a power of 2. Similar polynomial expressions hold for higher-rank groups, scaling with q raised to the dimension of the variety, modulated by factors from the Borel subgroup and Weyl group order. These formulas underscore the groups' non-abelian nature and their role as rich examples beyond cyclic or abelian structures. In the classification of finite simple groups, groups of Lie type constitute the largest infinite families, comprising 16 series (including twists) that account for the majority of all known non-abelian simple groups, with the remainder being alternating groups, cyclic groups of prime order, and 26 sporadics. This prominence stems from their systematic construction, which unifies diverse linear and orthogonal groups under algebraic geometry. The theory originated with Claude Chevalley's 1955 construction of the untwisted groups via integral forms of Lie algebras, later extended by Robert Steinberg in the late 1950s through twisting mechanisms to encompass the full spectrum of Lie-type simples.[34]Sylow theory
Sylow theorems
A Sylow p-subgroup of a finite group G is a maximal p-subgroup of G, meaning a subgroup P \leq G whose order is p^k, where p^k is the highest power of the prime p dividing |G|.[35] Such subgroups play a central role in the structure theory of finite groups, generalizing the concept of p-groups to arbitrary finite groups.[36] The first Sylow theorem guarantees the existence of such subgroups. Sylow's first theorem states that for a finite group G and a prime p dividing |G|, there exists at least one Sylow p-subgroup of G. The proof proceeds by induction on the order of G, building larger p-subgroups step by step. If |G| is a power of p, then G itself is the Sylow p-subgroup. Otherwise, start with a nontrivial p-subgroup H and consider its action on the left cosets of itself by left multiplication. The fixed-point congruence implies that the normalizer N_G(H) properly contains H, allowing construction of a larger p-subgroup by induction or Cauchy's theorem.[35] Continuing this process yields a maximal p-subgroup of order p^k.[37] The second Sylow theorem addresses the conjugacy of these subgroups. Sylow's second theorem asserts that any two Sylow p-subgroups of G are conjugate in G, and moreover, the number n_p of distinct Sylow p-subgroups satisfies n_p \equiv 1 \pmod{p} and divides |G|/p^k. The conjugacy follows from the transitivity of the conjugation action of G on the set of Sylow p-subgroups. For a fixed Sylow p-subgroup P, the stabilizer under this action is N_G(P), so n_p = [G : N_G(P)], which divides |G|/p^k by Lagrange's theorem (since |N_G(P)| is divisible by p^k). The congruence n_p \equiv 1 \pmod{p} arises from the action of P on the left cosets of N_G(P) by left multiplication: since P \leq N_G(P), this action fixes exactly one coset (the trivial one), and fixed-point theorems imply the number of cosets (i.e., n_p) is congruent to 1 modulo p.[35][36] Finally, Sylow's third theorem provides a criterion for normality: a Sylow p-subgroup P of G is normal in G if and only if it is the unique Sylow p-subgroup of G. The forward direction follows immediately from the second theorem, as conjugates of a normal subgroup are itself. Conversely, if P is unique, then it is fixed by all conjugations, hence normal.[37] These theorems, originally proved by Peter Ludvig Sylow in 1872, form the foundation for much of modern finite group theory.Applications of Sylow theorems
The Sylow theorems provide powerful tools for classifying finite groups of prime-power product order, particularly when the order is pq with distinct primes p < q. In such a group G, the number of Sylow q-subgroups n_q divides p and satisfies n_q \equiv 1 \pmod{q}. Since p < q, the only possibility is n_q = 1, so the Sylow q-subgroup is unique and hence normal in G.[38] The number of Sylow p-subgroups n_p divides q and satisfies n_p \equiv 1 \pmod{p}, so n_p = 1 or q. The case n_p = q occurs if and only if q \equiv 1 \pmod{p}, or equivalently, p divides q-1. If n_p = 1, then both Sylow subgroups are normal, and G is cyclic of order pq. If n_p = q > 1, then G is a non-trivial semidirect product of its normal Sylow q-subgroup by the Sylow p-subgroup.[38][39] A key consequence of the Sylow theorems concerns solvability: if every Sylow subgroup of a finite group G is normal, then G is the direct product of its Sylow subgroups. This follows because the Sylow subgroups pairwise normalize each other, and their elements commute across distinct primes, yielding a direct decomposition into cyclic p-groups for each prime p dividing |G|.[38] Such groups are necessarily solvable, as the direct product of solvable groups (here, cyclic p-groups) is solvable. The Sylow theorems facilitate this by confirming the uniqueness and normality of these subgroups via n_p = 1 for all primes p.[38] The classification of groups of order 12 illustrates practical applications of Sylow counts to determine group structure. For |G| = 12 = 2^2 \cdot 3, the possible values are n_3 = 1 or $4 (dividing 4 and \equiv 1 \pmod{3}) and n_2 = 1 or $3 (dividing 3 and \equiv 1 \pmod{2}). If n_3 = 1, the normal Sylow 3-subgroup Q \cong \mathbb{Z}_3 admits either a direct product (yielding \mathbb{Z}_{12} or \mathbb{Z}_6 \times \mathbb{Z}_2) or a semidirect product by a Sylow 2-subgroup (yielding the dihedral group D_{12} of order 12 or the dicyclic group of order 12). If n_3 = 4, then n_2 = 1 or $3; the case n_2 = 3 is impossible as it would imply more than 12 elements, so n_2 = 1 and G \cong A_4, the alternating group on 4 letters. These cases exhaust the five isomorphism classes of groups of order 12.[40] The Sylow theorems also transfer to the study of composition factors by helping identify normal subgroups and quotients in a composition series. For instance, a normal Sylow p-subgroup yields a quotient that is a p-complement, allowing recursive decomposition; non-trivial Sylow counts can signal non-solvability or specific simple factors, as in the classification of finite simple groups where Sylow subgroups constrain possible orders.[38] Burnside's normal p-complement theorem provides a criterion for the existence of a normal Hall p'-subgroup. Specifically, if G is finite and P is a Sylow p-subgroup with P \leq Z(N_G(P)), then G has a normal p-complement N (a normal Hall subgroup of order |G|/|P|) such that G = N \rtimes P. This condition leverages the Sylow theorems by ensuring the normalizer controls fusion and centralization within P, often verified via n_p \equiv 1 \pmod{p} and additional divisibility.[41] The theorem implies solvability in cases where such complements exist for the smallest prime p dividing |G|.[41]Group actions
Cayley's theorem
Cayley's theorem states that every finite group G is isomorphic to a subgroup of the symmetric group \Sym(G) consisting of all bijections from G to itself.[42][43] This isomorphism arises from the left regular action of G on itself, defined by g \cdot x = gx for all g, x \in G.[44][43] To establish this, consider the map \phi: G \to \Sym(G) given by \phi(g)(x) = gx. First, \phi(g) is a bijection for each g \in G: it is injective because if g x_1 = g x_2, then left multiplication by g^{-1} yields x_1 = x_2; it is surjective because for any x \in G, there exists x' = g^{-1} x such that \phi(g)(x') = x.[44] Next, \phi is a group homomorphism: \phi(g_1 g_2)(x) = (g_1 g_2) x = g_1 (g_2 x) = \phi(g_1)(\phi(g_2)(x)), so \phi(g_1 g_2) = \phi(g_1) \circ \phi(g_2).[44][43] Finally, \phi is injective (hence faithful), as \phi(g_1) = \phi(g_2) implies \phi(g_1)(e) = \phi(g_2)(e), so g_1 = g_2 where e is the identity.[44][43] Thus, \phi embeds G as a subgroup of \Sym(G), which has degree |G|.[44] The theorem implies that every finite group can be realized concretely as a permutation group acting regularly on a set of size equal to its order, providing a bridge from abstract algebraic structures to explicit symmetries of finite sets.[45] This realization underscores the connection between group theory and the study of symmetries, as permutation groups model transformations preserving set structure.[45] For example, consider the symmetric group S_3 of order 6, generated by a 3-cycle f = (1\ 2\ 3) and a transposition g = (1\ 2). The left regular action embeds S_3 into \Sym(S_3), where elements act by left multiplication on the group's own elements (listed as e, f, f^2, g, fg, f^2 g). For instance, \phi(f) permutes these as e \mapsto f, f \mapsto f^2, f^2 \mapsto e, g \mapsto fg, fg \mapsto f^2 g, f^2 g \mapsto g, corresponding to the cycle (e\ f\ f^2)(g\ fg\ f^2 g).[43] This permutation representation faithfully captures S_3's structure within the larger symmetric group of degree 6.[43]Burnside's lemma
In group theory, a finite group G acts on a finite set X if there is a map G \times X \to X, denoted (g, x) \mapsto g \cdot x, such that the identity element fixes every point and the action is compatible with the group operation: e \cdot x = x and (gh) \cdot x = g \cdot (h \cdot x) for all g, h \in G and x \in X. The orbit of an element x \in X is the set \{ g \cdot x \mid g \in G \}, which partitions X into equivalence classes under the relation x \sim y if y = g \cdot x for some g \in G. The stabilizer of x is the subgroup \operatorname{Stab}_G(x) = \{ g \in G \mid g \cdot x = x \}. Burnside's lemma provides a method to count the number of orbits in such an action. For a finite group G acting on a finite set X, the number of orbits is given by \frac{1}{|G|} \sum_{g \in G} |\operatorname{Fix}(g)|, where \operatorname{Fix}(g) = \{ x \in X \mid g \cdot x = x \} is the set of fixed points of g.[46] This formula, originally attributed to Frobenius but popularized by Burnside, arises from averaging the number of fixed points over all group elements.[47] To sketch the proof, consider the sum \sum_{g \in G} |\operatorname{Fix}(g)|, which equals \sum_{x \in X} |\operatorname{Stab}_G(x)| by double counting the pairs (g, x) with g \cdot x = x. For each orbit O, the stabilizers of its elements are equal, and by the orbit-stabilizer theorem, |O| = |G| / |\operatorname{Stab}_G(x)| for x \in O, so \sum_{x \in O} |\operatorname{Stab}_G(x)| = |O| \cdot |\operatorname{Stab}_G(x)| = |G|. Summing over all orbits thus yields \sum_{x \in X} |\operatorname{Stab}_G(x)| = |G| \cdot k, where k is the number of orbits, proving the lemma.[48] Burnside's lemma has key applications in finite group theory, such as counting necklaces under the action of the cyclic group of rotations, where elements with cycle structures matching the necklace's symmetries contribute to fixed colorings.[49] It also enumerates conjugacy classes in G by applying the lemma to the conjugation action on G itself, yielding the number of classes as \frac{1}{|G|} \sum_{g \in G} |\operatorname{C}_G(g)|, where \operatorname{C}_G(g) is the centralizer of g.[48] Additionally, it counts conjugacy classes of subgroups, providing the number of subgroups up to isomorphism under conjugation. A representative example is counting the number of distinct colorings of an n-element set with k colors up to permutation by the symmetric group S_n. The set X consists of all functions from \{1, \dots, n\} to \{1, \dots, k\}, with S_n acting by (g \cdot f)(i) = f(g^{-1} i). A permutation g fixes a coloring f if f is constant on the cycles of g, so |\operatorname{Fix}(g)| = k^{c(g)} where c(g) is the number of cycles in g. The number of orbits is thus \frac{1}{n!} \sum_{g \in S_n} k^{c(g)}.[48]Structure theorems
Direct and semidirect products
The direct product of two finite groups G and H, denoted G \times H, consists of ordered pairs (g, h) with g \in G and h \in H, equipped with the componentwise operation (g_1, h_1)(g_2, h_2) = (g_1 g_2, h_1 h_2).[50] This construction yields a group of order |G| \cdot |H|, and the projections onto each factor are surjective homomorphisms with kernels isomorphic to the other factor.[51] If both G and H are abelian, then G \times H is abelian, since for any (g_1, h_1), (g_2, h_2) \in G \times H, the commutator [(g_1, h_1), (g_2, h_2)] = ([g_1, g_2], [h_1, h_2]) = (e_G, e_H).[52] An internal direct product characterizes when a finite group G decomposes as such a product of its subgroups: G = N \times K if and only if N and K are normal subgroups of G, N \cap K = \{e\}, and N K = G.[50] In this case, every element of G uniquely writes as n k with n \in N and k \in K, and the multiplication follows the direct product rule.[51] A representative example is the Klein four-group, which is the direct product \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, consisting of elements of order dividing 2 under componentwise addition modulo 2.[53] The semidirect product provides a more general construction for building finite groups, incorporating a nontrivial action. Given finite groups N and H and a homomorphism \phi: H \to \Aut(N), the external semidirect product N \rtimes_\phi H has underlying set N \times H with operation (n_1, h_1)(n_2, h_2) = (n_1 \cdot \phi(h_1)(n_2), h_1 h_2).[54] This forms a group where N (identified with N \times \{e_H\}) is normal and H (identified with \{e_N\} \times H) is a subgroup, with N \cap H = \{e\} and N H = N \rtimes_\phi H.[55] Internally, G = N \rtimes H if N is normal in G, H is a subgroup, N \cap H = \{e\}, and N H = G, with conjugation in G inducing the action \phi.[54] A classic example is the symmetric group S_3, which is the semidirect product \mathbb{Z}/3\mathbb{Z} \rtimes \mathbb{Z}/2\mathbb{Z}, where \mathbb{Z}/2\mathbb{Z} acts on \mathbb{Z}/3\mathbb{Z} by inversion (the nontrivial automorphism sending $1 \mapsto 2 \pmod{3}).[54] Here, the order-3 rotation subgroup is normal, and the order-2 reflection complements it. The direct product arises as a special case of the semidirect product when \phi is the trivial homomorphism, yielding no twisting by automorphisms.[55]Solvable and nilpotent groups
A solvable group is a finite group G that possesses a subnormal series \{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_k = G such that each factor group G_{i+1}/G_i is abelian.[56] Equivalently, the derived series of G, defined by G^{(0)} = G and G^{(i+1)} = [G^{(i)}, G^{(i)}] (the commutator subgroup), terminates at the trivial subgroup after finitely many steps, i.e., G^{(n)} = \{e\} for some n.[56] This property captures groups that can be "built up" from abelian groups through extensions, reflecting a hierarchical structure amenable to inductive analysis.[57] All abelian groups are solvable, as their derived subgroup is trivial.[56] For instance, the symmetric group S_3 of order 6 is solvable, with derived series S_3 \triangleright A_3 \triangleright \{e\}, where A_3 is cyclic of order 3.[58] In contrast, the alternating group A_5 of order 60 is not solvable, as its derived subgroup equals itself, preventing the series from reaching the trivial group.[56] A finite group is nilpotent if its lower central series, defined by \gamma_1(G) = G and \gamma_{i+1}(G) = [G, \gamma_i(G)], terminates at the trivial subgroup, i.e., \gamma_m(G) = \{e\} for some m.[59] For finite groups, this is equivalent to the group being the direct product of its Sylow subgroups, each of which is normal.[59] Nilpotent groups form a subclass of solvable groups, as the lower central series refines to a subnormal series with abelian factors.[59] Every finite p-group is nilpotent (and hence solvable), since the center of a nontrivial finite p-group is nontrivial, allowing the upper central series to ascend to the whole group in finitely many steps.[59] Abelian groups are nilpotent of class 1, with trivial lower central series beyond the first term.[59] The group S_3 is solvable but not nilpotent, as its lower central series stabilizes at A_3 \neq \{e\}.[58] Burnside's normal p-complement theorem provides a criterion for solvability: if P is a Sylow p-subgroup of a finite group G such that P lies in the center of its normalizer N_G(P), then G has a normal p-complement (a normal Hall subgroup whose order is coprime to p and intersects P trivially).[60] Iteratively applying this theorem to the factors can establish solvability, as the existence of such complements reduces the problem to smaller solvable pieces.[60]Composition series and Jordan–Hölder theorem
A composition series of a finite group G is a finite chain of subgroups $1 = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G such that each quotient G_{i+1}/G_i is a simple group for $0 \leq i < n; these quotients are called the composition factors of the series.[61] Every finite group possesses at least one composition series, which can be constructed by iteratively selecting maximal normal subgroups until reaching the trivial subgroup.[62] The Jordan–Hölder theorem asserts that any two composition series of a finite group G have the same length n and the same composition factors up to isomorphism and permutation.[61] This uniqueness implies that the multiset of composition factors is an invariant of the group, providing a canonical decomposition into simple building blocks. The proof relies on the Schreier refinement theorem, which states that any two subnormal series of a group admit refinements that are equivalent, meaning their factor groups are isomorphic up to permutation and repetition.[63] To apply this to composition series, one refines both series using the Zassenhaus lemma to ensure maximal subnormal steps with simple factors, then removes isomorphic repetitions to match the factors pairwise; the process uses induction on the group order to handle the base case of simple groups.[63] A related concept is the chief series, a maximal chain of normal subgroups $1 = N_0 \trianglelefteq N_1 \trianglelefteq \cdots \trianglelefteq N_r = G where each N_{i+1}/N_i is a minimal normal subgroup of G/N_i, known as chief factors; unlike composition factors, chief factors need not be simple but are characteristically simple, often direct products of isomorphic simple groups.[64] The Jordan–Hölder theorem extends analogously to chief series, ensuring their factors are unique up to isomorphism and permutation.[64] For example, the symmetric group S_4 has chief series \{e\} \trianglelefteq V_4 \trianglelefteq A_4 \trianglelefteq S_4, where V_4 is the Klein four-group, with chief factors \mathbb{Z}_2 \times \mathbb{Z}_2, \mathbb{Z}_3, and \mathbb{Z}_2.[65] A corresponding composition series refines the nonsimple chief factor: \{e\} \trianglelefteq \langle (1\,2)(3\,4) \rangle \trianglelefteq V_4 \trianglelefteq A_4 \trianglelefteq S_4, yielding simple factors \mathbb{Z}_2, \mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_2.[65] In solvable groups, all composition factors are abelian, specifically cyclic of prime order.[61]Simple groups
Definition and basic properties
In group theory, a finite simple group is defined as a nontrivial finite group that possesses no normal subgroups other than the trivial subgroup and the group itself.[66] This definition, originally proposed by Évariste Galois, captures groups that cannot be decomposed nontrivially via normal subgroups, making them the "atoms" of finite group structure.[66] For abelian groups, simplicity implies that the group is cyclic of prime order. Specifically, if G is an abelian simple group, then |G| = p for some prime p, and G \cong \mathbb{Z}/p\mathbb{Z}.[67] This follows from the fact that any proper nontrivial subgroup of an abelian group is normal, so simplicity requires no such subgroups, which occurs precisely when the order is prime.[67] Non-abelian simple groups, by contrast, are infinite in number and include examples like the alternating group A_n for n \geq 5, which is simple because any normal subgroup must contain 3-cycles and thus generate the entire group.[23] A key property of simple groups is their role in composition series: every finite group has a composition series where the successive quotients (composition factors) are simple groups, and by the Jordan–Hölder theorem, these factors are unique up to isomorphism and ordering.[68] For a simple group G, the only maximal normal subgroup is the trivial subgroup \{e\} (since G itself is normal in G), emphasizing its indecomposability.[69] Although all known non-abelian finite simple groups have even order—with A_5 (order 60) as the smallest example—early conjectures sometimes questioned this, but counterexamples like A_5 confirm their existence.[66] Regarding automorphisms, the outer automorphism group \mathrm{Out}(G) = \mathrm{Aut}(G)/\mathrm{Inn}(G) measures symmetries beyond inner ones induced by conjugation; for most finite simple groups, \mathrm{Out}(G) is small, often of order 1 or 2.[70] The Schur multiplier M(G) = H_2(G, \mathbb{Z}) of a finite simple group G is the kernel of the universal central extension and is typically small or trivial for non-abelian cases; for instance, it is trivial for alternating groups A_n (n \geq 5) and cyclic groups \mathbb{Z}_p.[71] This multiplier encodes stem extensions and has been computed for all known finite simple groups, aiding their classification.[71]Feit–Thompson theorem
The Feit–Thompson theorem states that every finite group of odd order is solvable. This result, also known as the odd order theorem, was established by Walter Feit and John G. Thompson in their seminal 1963 paper "Solvability of groups of odd order," published in the Pacific Journal of Mathematics. The proof spans 255 pages and represents one of the most intricate arguments in finite group theory at the time. The proof strategy is divided into local and global components, relying heavily on advanced techniques from representation theory. The local analysis employs character theory of finite groups, particularly the use of transfers—maps that relate characters of a group to those of its subgroups—to investigate the structure of Sylow subgroups and detect nilpotency in certain formations. Formation theory, a framework for constructing groups via subnormal series with specified factor groups, is then applied in the global phase to show that a minimal counterexample must possess a normal solvable subgroup, leading to a contradiction. Subsequent simplifications, such as those by Bender in the 1970s, have reduced the length while preserving the core ideas of character-theoretic transfers and formations. A key corollary of the theorem is that all non-abelian simple finite groups have even order, since a non-abelian simple group of odd order would contradict solvability while violating simplicity. This implication was pivotal in the classification of finite simple groups, as it eliminated the need to consider odd-order candidates beyond cyclic groups of prime order, thereby focusing efforts on even-order cases and serving as the foundational step in the decades-long project completed in the 1980s and 2000s.Classification of finite simple groups
The Classification of Finite Simple Groups (CFSG) is one of the most significant achievements in modern mathematics, providing a complete enumeration of all finite simple groups up to isomorphism. This theorem asserts that every finite simple group falls into one of four categories: cyclic groups of prime order, alternating groups A_n for n \geq 5, groups of Lie type defined over finite fields, or one of 26 exceptional sporadic groups. The classification encompasses 18 infinite families in total (including the cyclic and alternating ones within the broader count) and these 26 sporadics, with no others existing.[72] The abelian simple groups are exactly the cyclic groups \mathbb{Z}_p where p is prime. The non-abelian infinite families consist of the alternating groups A_n (n \geq 5), which are the even permutations on n letters, and the 16 families of groups of Lie type. These Lie-type groups arise as finite analogues of Lie groups and include Chevalley groups such as the projective special linear groups \mathrm{PSL}(n, q), symplectic groups \mathrm{PSp}(2m, q), and exceptional types like E_8(q), all defined over the finite field \mathbb{F}_q where q is a prime power; twisted variants, such as the unitary groups \mathrm{PSU}(n, q), Suzuki groups \mathrm{Sz}(q) for q = 2^{2m+1}, and Ree groups {}^2G_2(q) or {}^2F_4(q).[68][72] The 26 sporadic simple groups are finite exceptions that do not belong to any infinite family and were discovered individually through various constructions. Notable examples include the Mathieu groups M_{11}, M_{12}, M_{22}, M_{23}, and M_{24}, which are highly symmetric permutation groups related to Steiner systems; the Janko groups J_1, J_2, J_3, and J_4; the Conway groups \mathrm{Co}_1, \mathrm{Co}_2, and \mathrm{Co}_3, linked to Leech lattice symmetries; and the Monster group \mathbb{M}, the largest sporadic with order $8,089,174,247,945,128,785,886,459,904,961,710,757,005,754,368,000,000,000 \approx 8 \times 10^{53}. Twenty of these sporadics are subquotients of the Monster (the "Happy Family"), while the remaining six are "pariahs" with no such connections.[73] The proof of the CFSG involved over 100 mathematicians and spanned more than 50 years, culminating in over 10,000 pages across hundreds of papers; it was initially announced as complete in 1983 by Daniel Gorenstein but required revisions, with the final gaps closed in 2004 by Michael Aschbacher and Stephen D. Smith. Ongoing projects, including a second-generation proof by Gorenstein, Lyons, and Solomon, aim to streamline and verify the result further. A key implication is that, by the Jordan–Hölder theorem, every finite group admits a composition series whose factors are these simple groups, allowing all finite groups to be understood as "built" from them via group extensions, direct products, and semidirect products.[72]Enumeration
Number of groups of order n
The number g(n) of groups of order n up to isomorphism, also denoted f(n) in some literature, counts the distinct isomorphism classes of finite groups with exactly n elements. This function is multiplicative in a certain sense but highly irregular, with g(n) = 1 for all n \leq 3 (the trivial group for n=1, and the cyclic groups \mathbb{Z}/2\mathbb{Z} and \mathbb{Z}/3\mathbb{Z} for n=2,3), and it grows rapidly thereafter, particularly when n is highly composite, reflecting the increasing complexity of group structures as more prime factors are introduced. For instance, the proliferation arises from combinations of Sylow subgroups and extensions, leading to an explosion in possibilities for orders with many small prime factors. For specific cases, explicit formulas exist. When n = p^k is a prime power, the enumeration of p-groups of order p^k is a central problem, with the asymptotic growth given by Higman's formula: g(p^k) = p^{\frac{2}{27} k^3 + O(k^{5/2})}. This reflects the polynomial-in-p nature of the count in the exponent of k, driven by the variety of nilpotent structures and relations in p-groups. For the subclass of abelian groups of order n, the fundamental theorem of finite abelian groups provides a complete classification up to isomorphism via invariant factors or elementary divisors, yielding g_{\text{abelian}}(n) = \prod_p p(k_p), where the product is over primes p dividing n, k_p = v_p(n) is the p-adic valuation, and p(m) denotes the partition function counting integer partitions of m. In general, no closed-form formula for g(n) exists, but computational methods enable determination for moderate n. Systems like the GAP (Groups, Algorithms, Programming) computer algebra package include the SmallGroups library, which catalogs all isomorphism classes of groups up to order 2000 (excluding orders 1024 and 1536 due to computational intensity), facilitating enumeration, identification, and structural analysis via algorithms for Sylow subgroups and presentations. Online databases built on such libraries, including those integrated with GAP, provide accessible lookups and verify isomorphisms for research. Asymptotically, bounds on g(n) capture the explosive growth without exact formulas. Pyber established an upper bound g(n) \leq n^{\left( \frac{2}{27} + o(1) \right) \mu(n)^2}, where \mu(n) = \max_p v_p(n) is the largest exponent in the prime factorization of n, implying \log g(n) < \left( \frac{2}{27} + o(1) \right) \mu(n)^2 \log n. These polynomial-exponential bounds highlight that g(n) is subexponential in n, with the dominant contribution often from p-groups for small primes like p=2, aligning with lower bounds from the Higman-Sims asymptotic that suggest \log g(n) grows on the order of (\log n)^3 for prime-power n.Groups of small order
The only group of order 1 is the trivial group. For a prime number p, there is exactly one group of order p up to isomorphism: the cyclic group \mathbb{Z}_p. Groups of order p^2, where p is prime, are all abelian; there are exactly two up to isomorphism: the cyclic group \mathbb{Z}_{p^2} and the elementary abelian group \mathbb{Z}_p \times \mathbb{Z}_p. For order pq with distinct primes p < q, the classification depends on the divisibility condition p \mid (q-1). If p does not divide q-1, the only group is the cyclic \mathbb{Z}_{pq}. If p divides q-1, there are exactly two groups: the cyclic \mathbb{Z}_{pq} and a non-abelian semidirect product \mathbb{Z}_q \rtimes \mathbb{Z}_p. This uses Sylow theorems to show the Sylow q-subgroup is normal and the action of \mathbb{Z}_p on it is determined by homomorphisms to \mathrm{Aut}(\mathbb{Z}_q) \cong \mathbb{Z}_{q-1}^\times. For example, for order 6 = 2 × 3 (where 2 divides 3-1), there are two groups: \mathbb{Z}_6 and S_3. Although order 12 = 2^2 × 3 is not of the form pq, there are five groups up to isomorphism: the abelian ones \mathbb{Z}_{12} and \mathbb{Z}_6 \times \mathbb{Z}_2; and the non-abelian ones A_4, the dihedral group D_{12} of order 12 (symmetries of regular hexagon), and the dicyclic group \mathrm{Dic}_3 (also known as the binary dihedral group of order 12). These are classified using Sylow subgroups and semidirect products, with the non-abelian examples arising from actions of Sylow 3-subgroups on Sylow 2-subgroups or vice versa. The numbers of groups of small order are tabulated below for n \leq 60, including the count of non-abelian groups. These enumerations stem from systematic computational constructions verifying all possibilities up to isomorphism.| Order n | Total groups | Non-abelian groups |
|---|---|---|
| 1 | 1 | 0 |
| 2 | 1 | 0 |
| 3 | 1 | 0 |
| 4 | 2 | 0 |
| 5 | 1 | 0 |
| 6 | 2 | 1 |
| 7 | 1 | 0 |
| 8 | 5 | 2 |
| 9 | 2 | 0 |
| 10 | 2 | 1 |
| 11 | 1 | 0 |
| 12 | 5 | 3 |
| 13 | 1 | 0 |
| 14 | 2 | 1 |
| 15 | 1 | 0 |
| 16 | 14 | 9 |
| 17 | 1 | 0 |
| 18 | 5 | 3 |
| 19 | 1 | 0 |
| 20 | 5 | 3 |
| 21 | 2 | 1 |
| 22 | 2 | 1 |
| 23 | 1 | 0 |
| 24 | 15 | 12 |
| 25 | 2 | 0 |
| 26 | 2 | 1 |
| 27 | 5 | 2 |
| 28 | 4 | 2 |
| 29 | 1 | 0 |
| 30 | 4 | 3 |
| 31 | 1 | 0 |
| 32 | 51 | 44 |
| 33 | 1 | 0 |
| 34 | 2 | 1 |
| 35 | 1 | 0 |
| 36 | 14 | 10 |
| 37 | 1 | 0 |
| 38 | 2 | 1 |
| 39 | 2 | 1 |
| 40 | 14 | 11 |
| 41 | 1 | 0 |
| 42 | 6 | 5 |
| 43 | 1 | 0 |
| 44 | 4 | 2 |
| 45 | 2 | 0 |
| 46 | 2 | 1 |
| 47 | 1 | 0 |
| 48 | 52 | 47 |
| 49 | 2 | 0 |
| 50 | 5 | 3 |
| 51 | 1 | 0 |
| 52 | 9 | 6 |
| 53 | 1 | 0 |
| 54 | 15 | 12 |
| 55 | 2 | 1 |
| 56 | 13 | 10 |
| 57 | 2 | 1 |
| 58 | 2 | 1 |
| 59 | 1 | 0 |
| 60 | 13 | 11 |