List of finite simple groups
The list of finite simple groups comprises all finite groups that are simple—meaning they possess no nontrivial normal subgroups—and are thus the basic building blocks of the theory of finite groups, analogous to prime numbers in the arithmetic of the integers.[1] The Classification of Finite Simple Groups (CFSG), a monumental theorem in mathematics, asserts that every finite simple group is isomorphic to exactly one member of either one of 18 infinite families or one of 26 exceptional sporadic groups, with the proof spanning over 10,000 pages across hundreds of articles by more than 100 mathematicians and finalized in 2004 after initial completion in 1983.[2][1] The infinite families consist of: (1) the cyclic groups \mathbb{Z}_p of prime order p, which are the only abelian simple groups; (2) the alternating groups A_n for n \geq 5, consisting of all even permutations of n elements; and (3) the 16 families of groups of Lie type, which arise as certain matrix groups over finite fields and include examples like the projective special linear groups \mathrm{PSL}_n(q), Suzuki groups, and Ree groups.[2] These families account for the vast majority of finite simple groups, with orders growing indefinitely as parameters like n and q increase.[1] The 26 sporadic groups, by contrast, do not fit into these families and include notable examples such as the Mathieu groups (the smallest sporadics, discovered in the 19th century), the Monster group (the largest, with order approximately $8 \times 10^{53}), and others like the Baby Monster and Harada-Norton group.[2] These sporadics were identified through exhaustive searches and theoretical work during the classification effort, with 20 of them forming a "happy family" related to subquotients of the Monster and the remaining 6 as "pariahs."[2] The complete enumeration provided by the CFSG enables the explicit listing of all finite simple groups up to isomorphism, facilitating applications in areas such as representation theory, combinatorics, and symmetry studies in physics.[1]Overview
Definition and properties
A finite simple group is defined as a nontrivial finite group that possesses no nontrivial proper normal subgroups, meaning its only normal subgroups are the trivial subgroup and the group itself.[3] This property distinguishes simple groups as the "atoms" or indecomposable building blocks within the structure of finite group theory, as they cannot be broken down further via normal subgroups.[4] Key properties of finite simple groups include their role as composition factors in the Jordan-Hölder decomposition of any finite group. The Jordan-Hölder theorem states that every finite group admits a composition series—a finite chain of subgroups where each factor group is simple—and that any two such series have the same length and isomorphic composition factors up to permutation.[5] Thus, the simple groups serve as the fundamental constituents from which all finite groups are constructed through group extensions. For instance, simplicity ensures that these groups cannot be factored nontrivially, making them essential in understanding the solvability and structure of more complex groups like symmetric groups, which arise as extensions involving simple components.[6] To illustrate the significance of this definition, consider the symmetric group S_3, which is non-simple because it contains the alternating subgroup A_3, a normal subgroup of order 3 isomorphic to the cyclic group \mathbb{Z}_3.[7] In contrast, simple groups lack such intermediate normal structure. The classification theorem reveals that every finite simple group falls into one of four categories: cyclic of prime order, alternating, of Lie type, or sporadic, though the full proof spans thousands of pages.[2]Classification theorem
The classification theorem of finite simple groups, often called the enormous theorem, asserts that every finite simple group is isomorphic to one of the following: a cyclic group of prime order, an alternating group A_n for n \geq 5, a group of Lie type over a finite field, or one of 26 sporadic groups. This classification encompasses 16 infinite families of groups of Lie type (including classical and exceptional types, along with their twisted variants) plus the two infinite families of cyclic and alternating groups, with the sporadics forming a finite exceptional list.[8] The historical pursuit of this theorem traces back to William Burnside's early 20th-century program, which used character theory to establish that every non-abelian finite simple group has even order, laying groundwork for systematic classification efforts. Major advances occurred in the mid-20th century, with the 1963 Feit-Thompson theorem proving that groups of odd order are solvable, thereby restricting non-abelian simple groups to even order and enabling focus on 2-local structures. Subsequent contributions from researchers including Douglas Goldschmidt, Michael Aschbacher, and others in the 1970s and 1980s addressed subgroup structures and case analyses; Daniel Gorenstein announced near-completion in 1983, but gaps persisted until Aschbacher and Stephen Smith filled the final quasithin case in 2004, with minor revisions thereafter. The proof strategy reduces arbitrary finite simple groups to known forms through analysis of local structures, often starting with centralizers of involutions or elements of prime order to identify components or force recognition as a known type. Key techniques include examining cases in characteristics 2 and 3 separately, employing signalizer functors to control p-local subgroups and eliminate potential normal solvable cores, and using Brauer's method to decompose centralizers into quasiprimitive components that match Lie-type or other classified forms. While the full proof spans thousands of pages across hundreds of papers, these strategies emphasize iterative reduction via uniqueness theorems for centralizers and subgroup lattices, without a single overarching argument. The theorem implies that all finite simple groups are uniquely determined up to isomorphism within their families, with dedicated uniqueness cases ensuring no unclassified exceptions arise.[9] Moreover, the infinite families—such as alternating groups of arbitrarily large degree and Lie-type groups over expanding finite fields—establish that there are infinitely many finite simple groups in total, contrasting with the finite sporadic collection.[9]Cyclic simple groups
Structure and examples
Cyclic simple groups are precisely the cyclic groups of prime order, denoted \mathbb{Z}_p or C_p, where p is a prime number. These groups consist of p elements and are generated by any single non-identity element, which has order exactly p.[10][11] As a consequence of their prime order, they possess no proper nontrivial subgroups, ensuring their simplicity.[10] Representative examples include \mathbb{Z}_2, the cyclic group of order 2, which is isomorphic to the symmetry group of a single reflection; \mathbb{Z}_3, of order 3, familiar as the rotational symmetries of an equilateral triangle; and \mathbb{Z}_5, of order 5, extending to larger primes such as 7, 11, or arbitrarily large ones, with no upper bound on their orders.[10][11] These groups are abelian, meaning their operation is commutative, and in any abelian group, every subgroup is normal. However, the simplicity condition—no nontrivial normal subgroups—restricts finite abelian groups to prime order, as any composite order would admit a proper normal subgroup of index greater than 1.[10][11] Consequently, any nontrivial homomorphism from \mathbb{Z}_p has image isomorphic to \mathbb{Z}_p itself, with no proper nontrivial quotients.[10] The structure of \mathbb{Z}_p can be represented concretely as the additive group of integers modulo p, where elements are equivalence classes {{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}}, {{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}}, \dots, [p-1] under addition, generated by {{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}}.[10][11]Orders and generation
The cyclic simple groups are precisely the cyclic groups \mathbb{Z}_p of prime order p, where the order of the group equals p. These groups form the abelian component of the classification of finite simple groups, distinguished as the only abelian simple groups. The smallest such group is \mathbb{Z}_2 of order 2, followed by \mathbb{Z}_3 of order 3, \mathbb{Z}_5 of order 5, \mathbb{Z}_7 of order 7, \mathbb{Z}_{11} of order 11, and so forth, with the sequence of orders corresponding directly to the sequence of prime numbers.[10] There are infinitely many cyclic simple groups, as there are infinitely many prime numbers, a fact established by Euclid's classical proof that assumes a finite list of primes and constructs a new prime via the product of those primes plus one.[12] Unlike other families of finite simple groups, which have bounded or more complex order formulas, the cyclic simple groups have no upper bound on their orders, reflecting the unbounded nature of the primes.[12] Each cyclic simple group \mathbb{Z}_p is singly generated: it is the cyclic subgroup \langle g \rangle for any non-identity element g, satisfying the relation g^p = e, where e is the identity.[10] More precisely, every group of prime order is cyclic, and thus generated by any of its p-1 non-identity elements.[13] An explicit construction embeds \mathbb{Z}_p as the subgroup generated by a p-cycle in the symmetric group S_p, such as \langle (1\ 2\ \dots\ p) \rangle, which yields a faithful permutation representation of order p.[10] This generation highlights the minimal structure of these groups, requiring just one generator and one defining relation.Alternating simple groups
Definition and isomorphism
The alternating group A_n on n letters is defined as the kernel of the sign homomorphism \sgn: S_n \to \mathbb{Z}/2\mathbb{Z}, where S_n is the symmetric group on n letters; thus, A_n consists precisely of the even permutations in S_n.[14] As the kernel of a group homomorphism, A_n is a normal subgroup of S_n of index 2.[14] The alternating groups A_n are simple for all n \geq 5, a result first established by Camille Jordan.[15][16] In contrast, A_4 is not simple, as it contains the Klein four-group \{ e, (12)(34), (13)(24), (14)(23) \} as a proper normal subgroup of order 4.[15] The group A_5 is thus the smallest non-abelian simple alternating group.[15] Alternating groups A_n and A_m are isomorphic if and only if n = m, since they have distinct orders |A_n| = n!/2 for n \neq m.[14] Moreover, no alternating group is isomorphic to any simple group from other families in the classification of finite simple groups.[17] The automorphism group \Aut(A_n) is isomorphic to S_n for n \neq 6, with the outer automorphisms arising from conjugation by elements of S_n \setminus A_n.[17] For n = 6, \Aut(A_6) is larger, with outer automorphism group isomorphic to \mathbb{Z}_2 \times \mathbb{Z}_2.[17]Smallest examples and orders
The alternating groups A_n for n \geq 5 provide the smallest non-abelian simple groups in the classification, starting with A_5 of order 60, followed by A_6 of order 360, A_7 of order 2520, and A_8 of order 20160.[18] The general order of the alternating group is given by the formula |A_n| = n!/2 for n \geq 2, reflecting that it is the kernel of the sign homomorphism from the symmetric group S_n to \mathbb{Z}/2\mathbb{Z}./04%3A_Families_of_Groups/4.04%3A_Alternating_Groups) Among these, A_5 is particularly notable for its realizations as the group of rotational symmetries of the regular icosahedron (or dodecahedron) and as the projective special linear group \mathrm{[PSL](/page/PSL)}(2,5).[19]_is_isomorphic_to_A5) These isomorphisms highlight A_5's role in both geometric and linear algebraic contexts, though projective representations are not detailed here. The following table lists the orders of the smallest alternating simple groups:| Group | Order |
|---|---|
| A_5 | 60 |
| A_6 | 360 |
| A_7 | 2520 |
| A_8 | 20160 |
| A_9 | 181440 |
| A_{10} | 1814400 |
Groups of Lie type
Classical Chevalley groups
The classical Chevalley groups are the finite simple groups of Lie type arising from the classical irreducible root systems of types A_n, B_n, C_n, and D_n (n \geq 1), constructed uniformly over the finite field \mathbb{F}_q where q is a power of a prime. Introduced by Claude Chevalley in 1955, these groups are generated by root elements corresponding to the roots of the associated semisimple Lie algebra, with the simply connected form quotiented by its center to yield the simple adjoint form under appropriate conditions on n and q. They form infinite families within the classification of finite simple groups, alongside alternating and sporadic groups, and capture linear, unitary, symplectic, and orthogonal structures in finite geometry. The group A_n(q) is isomorphic to the projective special linear group \mathrm{PSL}_{n+1}(q), the group of (n+1) \times (n+1) matrices over \mathbb{F}_q of determinant $1$, modulo the scalar matrices. Its order is given by |A_n(q)| = \frac{1}{d} \, q^{n(n+1)/2} \prod_{i=1}^n (q^{i+1} - 1), where d = \gcd(n+1, q-1). This group is simple for all n \geq 1 except the cases A_1(2) \cong S_3 and A_1(3) \cong A_4. A representative example is \mathrm{PSL}_2(q) for q \geq 4, which is simple and includes groups like \mathrm{PSL}_2(5) \cong A_5 of order $60. Larger instances, such as \mathrm{PSL}_3(2) of order $168, illustrate the group's role in generating projective geometries over finite fields.[21] For type B_n(q) with n > 1, the group is the projective special odd orthogonal group \mathrm{PSO}_{2n+1}(q), preserving a non-degenerate quadratic form of maximal Witt index n on a (2n+1)-dimensional vector space over \mathbb{F}_q. The order formula is |B_n(q)| = \frac{1}{d} \, q^{n^2} \prod_{i=1}^n (q^{2i} - 1), with d = \gcd(2, q-1). These groups are simple for n \geq 2 except for B_2(2) \cong S_6, which is not simple; the simple groups of type B_n(q) arise for q \geq 3 or n \geq 3. An example is \mathrm{SO}_5(3) \cong \mathrm{PSO}_5(3) of order $10800$, which embeds into the automorphism group of certain block designs.[21] The type C_n(q) for n > 2 corresponds to the projective symplectic group \mathrm{PSp}_{2n}(q), the simple quotient of the symplectic group \mathrm{Sp}_{2n}(q) that preserves a non-degenerate alternating bilinear form on a $2n-dimensional space over \mathbb{F}_q$. Its order is |C_n(q)| = \frac{1}{d} \, q^{n^2} \prod_{i=1}^n (q^{2i} - 1), where d = \gcd(2, q-1), matching the order of B_n(q). The group is simple for n \geq 3 and for n=2, q \geq 3; the exception is \mathrm{PSp}_4(2) \cong S_6, which is not simple. A key example is \mathrm{PSp}_6(2) of order $1451520, which appears in the study of finite symplectic geometries and has applications in [coding theory](/page/Coding_theory). Note that B_n(q)andC_n(q)share the same order formula but differ in their geometric realizations, withC_n$ emphasizing symplectic structures.[21] Finally, the type D_n(q) for n > 3 is the projective special even orthogonal group \mathrm{PSO}_{2n}^+(q), preserving a quadratic form of Witt index n on a $2n-dimensional space over \mathbb{F}_q(the+$ superscript denotes the split form). The order is |D_n(q)| = \frac{1}{d} \, q^{n(n-1)} \prod_{i=1}^{n-1} (q^{2i} - 1) (q^n - 1), with d = \gcd(4, q^n - 1). These groups are simple for n \geq 4; exceptions occur for small n, such as D_2(q) \cong \mathrm{SL}_2(q) \times \mathrm{SL}_2(q) and D_3(q) \cong B_3(q). An illustrative case is \mathrm{PSO}_8^+(2) of order $33868800, which relates to the orthogonal geometry in eight dimensions and connects to the Leech lattice constructions in higher sporadic groups. The orthogonal groups in types B_nandD_n$ highlight the role of quadratic forms in distinguishing the classical series.[21]Exceptional Chevalley groups
The exceptional Chevalley groups are finite groups of Lie type arising from the exceptional irreducible root systems of types G_2, F_4, E_6, E_7, and E_8. These root systems consist of 12, 48, 72, 126, and 240 roots, respectively, with corresponding ranks 2, 4, 6, 7, and 8.[22] The groups, denoted G_2(q), F_4(q), E_6(q), E_7(q), and E_8(q), are defined over the finite field \mathbb{F}_q where q is a power of a prime, and they provide the untwisted forms associated with these root systems.[23] The orders of these groups (for the adjoint versions, which coincide with the simple groups except in cases with nontrivial center) are given by the following formulas:- |G_2(q)| = q^6 (q^6 - 1)(q^2 - 1)
- |F_4(q)| = q^{24} (q^{12} - 1)(q^8 - 1)(q^6 - 1)(q^2 - 1)
- |E_6(q)| = q^{36} (q^{12} - 1)(q^9 - 1)(q^8 - 1)(q^6 - 1)(q^5 - 1)(q^2 - 1)
- |E_7(q)| = q^{63} (q^{18} - 1)(q^{14} - 1)(q^{12} - 1)(q^{10} - 1)(q^8 - 1)(q^6 - 1)(q^2 - 1)
- |E_8(q)| = q^{120} (q^{30} - 1)(q^{24} - 1)(q^{20} - 1)(q^{18} - 1)(q^{14} - 1)(q^{12} - 1)(q^8 - 1)(q^2 - 1)
Twisted Chevalley groups
Twisted Chevalley groups form a class of finite simple groups of Lie type, obtained by twisting the universal Chevalley groups using a combination of the Frobenius endomorphism and a suitable graph automorphism of the underlying Dynkin diagram. The Frobenius map raises elements of the algebraic group over the finite field \mathbb{F}_{q^k} to the power q, where q is a prime power, and the graph automorphism has order 2 for types A_n, D_n, and E_6, or order 3 for type D_4. The resulting groups are the fixed-point subgroups under this twisting action, providing twisted analogs of the untwisted Chevalley groups while preserving simplicity under appropriate conditions. These constructions, developed in the context of algebraic groups, contribute essential families to the classification of finite simple groups.[24][17] The primary families are denoted ^2A_n(q), ^2D_n(q), ^2E_6(q), and ^3D_4(q), where q is a power of a prime and the superscripts indicate the twisting order. The group ^2A_n(q) is isomorphic to the projective special unitary group \mathrm{PSU}_{n+1}(q), arising from the order-2 automorphism of the A_n Dynkin diagram applied to the Chevalley group over \mathbb{F}_{q^2}. It is simple for n > 1 except small cases like ^2A_2(2) \cong U_3(2) \cong PSL_2(7). The order is |^2A_n(q)| = \frac{1}{d} q^{n(n+1)/2} \prod_{i=1}^n (q^{i+1} - (-1)^{i+1}), where d = \gcd(n+1, q+1). A representative example is ^2A_2(q) \cong \mathrm{PSU}_3(q), simple for q \geq 3.[21][17] The family ^2D_n(q) corresponds to the orthogonal group of minus type \Omega_{2n}^-(q), obtained similarly from the order-2 automorphism of the D_n Dynkin diagram on the group over \mathbb{F}_{q^2}. It is simple for n \geq 4 except small cases. The order formula is |^2D_n(q)| = \frac{q^{n(n-1)} (q^n + 1) \prod_{i=1}^{n-1} (q^{2i} - 1) \cdot 2}{d}, with d = \gcd(4, q^n + 1), adjusted for the simple version. For the exceptional types, ^2E_6(q) arises from twisting the E_6 diagram over \mathbb{F}_{q^2} and is simple for q \geq 2 (with minor exceptions in small characteristic); its order is |^2E_6(q)| = \frac{q^{36} (q^{18} + 1)(q^{12} - 1)(q^9 + 1)(q^8 - 1)(q^6 + 1)(q^5 - 1)(q^2 + 1)}{d}, with d = \gcd(3, q-1), emphasizing the complex interplay of cyclotomic factors. Finally, ^3D_4(q) uses the order-3 triality automorphism on the D_4 diagram over \mathbb{F}_{q^3} and is simple for q \geq 2; its order is |^3D_4(q)| = q^{12} (q^8 + q^4 + 1)(q^{12} - 1)(q^6 - 1), where the cubic twisting produces the irreducible factor q^8 + q^4 + 1. These orders generally follow the pattern adjusted for the specific Lie rank and twisting degree.[21][17]Suzuki and Ree groups
The Suzuki groups, denoted ^{2}B_{2}(q) or \mathrm{Sz}(q) where q=2^{2n+1} for integers n \geq 1, form an infinite family of non-abelian simple groups of Lie type defined over fields of characteristic 2. These groups arise as exceptional twisted Chevalley groups and were first constructed by Michio Suzuki in 1960 through their action on certain 4-dimensional vector spaces equipped with bilinear forms. The order of \mathrm{Sz}(q) is given by q^{2}(q^{2}+1)(q-1).[25] The smallest member is \mathrm{Sz}(8), with order $29120=2^{6} \cdot 5 \cdot 7 \cdot 13.[25] Their simplicity follows from the existence of a BN-pair structure, which allows reduction to the case of irreducible representations, combined with the Moufang properties of the associated buildings that ensure no nontrivial normal subgroups. Alternatively, simplicity can be established by verifying that these groups are perfect and act primitively on certain point sets with solvable stabilizers.[25] The Ree groups of type ^{2}G_{2}(q), often denoted \mathrm{Ree}(q) where q=3^{2n+1} for integers n \geq 1, constitute another family of twisted simple groups, this time in characteristic 3, and represent the small Ree series. Introduced by Reinhard Ree in 1960, they can be realized as automorphism groups of 7-dimensional vector spaces over \mathbb{F}_{q} with specific trilinear forms, generalizing the exceptional Lie algebra G_{2}. The order of \mathrm{Ree}(q) is (q^{3}+1)q^{3}(q-1).[25] The smallest simple example is \mathrm{Ree}(27), with order $10\,070\,400\,896 = 2^{3} \cdot 3^{6} \cdot 7 \cdot 13 \cdot 37 \cdot 56101.[25] Simplicity for these groups is proved using their BN-pair of rank 2, leveraging Tits' criterion that rules out proper normal subgroups via the irreducibility of the associated root system and the Moufang condition on parabolic subgroups. This approach exploits the thin chamber structure of the corresponding spherical building. Closely related is the Tits group, denoted ^{2}F_{4}(2)', which serves as the unique simple quotient of the twisted Chevalley group ^{2}F_{4}(2) by its center of order 3; it fits into the large Ree series as an exceptional case over characteristic 2. Constructed by Jacques Tits in 1964, this group has order $2^{11} \cdot 3^{3} \cdot 5^{2} \cdot 13 = 17\,971\,200. Its simplicity is established through the BN-pair framework, where the quotient inherits the necessary properties from the ambient group, ensuring no kernel beyond the center via Moufang building automorphisms and irreducibility arguments.Sporadic simple groups
Mathieu groups
The Mathieu groups comprise five sporadic finite simple groups: M_{11}, M_{12}, M_{22}, M_{23}, and M_{24}. These groups were the first sporadics to be identified and are notable for their connections to combinatorial designs and highly transitive actions. Their orders are as follows:| Group | Order |
|---|---|
| M_{11} | 7,920 |
| M_{12} | 95,040 |
| M_{22} | 443,520 |
| M_{23} | 10,200,960 |
| M_{24} | 244,823,040 |
- M_{11} is the automorphism group of S(4,5,11),
- M_{12} of S(5,6,12),
- M_{22} of S(3,6,22),
- M_{23} of S(4,7,23),
- M_{24} of S(5,8,24).
Janko and related groups
The Janko groups comprise four sporadic simple groups discovered sequentially by the Croatian mathematician Zvonimir Janko between 1965 and 1975, each postulated based on distinctive properties of involution centralizers and character degrees that set them apart from known simple groups.[30] These properties ensured their uniqueness within the classification of finite simple groups, where they appear as outliers not fitting into infinite families of alternating, Lie-type, or cyclic groups.[31] The first Janko group J_1 has order $2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 = 175560.[32] Janko postulated its existence in 1965 while studying groups with abelian Sylow $2-subgroups of order $8, identifying it as a simple group in which every involution has centralizer of structure $19:18 \times \mathrm{SL}(2,5).[33] Its construction followed from computational verification of the character table and simplicity proofs using modular representation theory. The second Janko group J_2 has order $2^7 \cdot 3^3 \cdot 5^2 \cdot 7 = 604800.[34] Predicted by Janko in 1966 alongside J_1 and J_3, it arises as a subgroup of the McLaughlin group \mathrm{McL} and features involutions with centralizer $2^{1+6} : (3 \cdot \mathrm{SL}(3,2)).[30] Marshall Hall provided an explicit construction in 1968 via a presentation and permutation representation, with uniqueness confirmed through subgroup analysis in \mathrm{McL}. The third Janko group J_3 has order $2^{10} \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \cdot 23 = 50232960.[34] Janko announced it in 1968 as a simple group with a single conjugacy class of involutions, each having centralizer $2^5 : \mathrm{SL}(2,11), and predicted its character degrees.[33] It admits a faithful primitive $3-transitive permutation representation of degree $37, and its construction was completed by Donald Higman and John McKay in 1972 using geometric methods involving quadratic forms. The fourth Janko group J_4, the largest sporadic simple group outside the Monster series, has order $2^{21} \cdot 3^3 \cdot 5 \cdot 7 \cdot 11^3 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 43 = 86775571046077562880.[32] Janko proposed it in 1975 based on a hypothetical simple group with involution centralizer $2^{11} : (\mathrm{SL}(2,11) \times \mathrm{SL}(2,5)) and specific character degrees, including a unique minimal non-trivial degree of $1333.[](https://www.semanticscholar.org/paper/Some-new-simple-groups-of-finite-order-Janko/f540c216c5904ed306bd5c84c0bbebf7a7af4fb0) It possesses a $3-transitive permutation action, and Robert Benson, John Conway, Simon Norton, Richard Parker, and Robert Thackray constructed it in 1978 as a subgroup of the symmetric group on $1333$ points via computational generation and verification of relations.[35]Conway and Fischer groups
The Conway groups, denoted \mathrm{Co}_1, \mathrm{Co}_2, and \mathrm{Co}_3, form a family of three sporadic simple groups discovered by John Horton Conway in the late 1960s through investigations into the symmetries of the Leech lattice, a highly symmetric 24-dimensional even unimodular lattice. The largest of these, \mathrm{Co}_1, arises as the quotient of the full automorphism group of the Leech lattice by its center \{\pm I\}, yielding a simple group of order $2^{21} \cdot 3^9 \cdot 5^4 \cdot 7^2 \cdot 11 \cdot 13 \cdot 23 \approx 8 \times 10^{24}. This group acts as the automorphism group preserving the lattice structure, highlighting its deep connection to lattice theory and coding in high dimensions. The groups \mathrm{Co}_2 and \mathrm{Co}_3 emerge as stabilizers of certain geometric objects within the action of \mathrm{Co}_1, with orders $2^{18} \cdot 3^6 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23 \approx 4 \times 10^{13} and $2^{10} \cdot 3^7 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23 \approx 5 \times 10^{11}, respectively. These subgroups retain significant symmetry properties inherited from \mathrm{Co}_1, including representations tied to the Leech lattice's orthogonal complements.[36][37][38][39] The Fischer groups, \mathrm{Fi}_{22}, \mathrm{Fi}_{23}, and \mathrm{Fi}_{24}', constitute another trio of sporadic simple groups identified by Bernd Fischer in the 1970s as part of his classification of groups generated by classes of 3-transpositions—involutions whose products of three distinct elements have order 2—using the Fischer-Clifford matrix method for computing induced representations. This approach, rooted in Clifford theory for modular representations, revealed these groups as exceptional cases beyond the known families of Lie type. \mathrm{Fi}_{22} has order $2^{17} \cdot 3^9 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \approx 6 \times 10^{17} and acts 3-transitively on a set of 3510 points, corresponding to the cosets of a maximal subgroup isomorphic to \mathrm{U}_6(2):2. Similarly, \mathrm{Fi}_{23} possesses order $2^{18} \cdot 3^{13} \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 23 \approx 4 \times 10^{20}, while \mathrm{Fi}_{24}', the simple subgroup of index 2 in the larger \mathrm{Fi}_{24}, has order $2^{21} \cdot 3^{16} \cdot 5^2 \cdot 7^3 \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 29 \approx 1 \times 10^{25} and exhibits 3-transitivity on 4096 points. These actions underscore the groups' high degree of symmetry in permutation representations.[40][41][42][39]| Group | Order (exact) | Approximate Order | Key Feature |
|---|---|---|---|
| \mathrm{Co}_1 | $2^{21} \cdot 3^9 \cdot 5^4 \cdot 7^2 \cdot 11 \cdot 13 \cdot 23 | $8 \times 10^{24} | Automorphism of Leech lattice modulo center |
| \mathrm{Co}_2 | $2^{18} \cdot 3^6 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23 | $4 \times 10^{13} | Stabilizer in \mathrm{Co}_1 action |
| \mathrm{Co}_3 | $2^{10} \cdot 3^7 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23 | $5 \times 10^{11} | Stabilizer in \mathrm{Co}_1 action |
| \mathrm{Fi}_{22} | $2^{17} \cdot 3^9 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 | $6 \times 10^{17} | 3-transitive on 3510 points |
| \mathrm{Fi}_{23} | $2^{18} \cdot 3^{13} \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 23 | $4 \times 10^{20} | Generated by 3-transpositions |
| \mathrm{Fi}_{24}' | $2^{21} \cdot 3^{16} \cdot 5^2 \cdot 7^3 \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 29 | $1 \times 10^{25} | 3-transitive on 4096 points |
Other first-generation sporadics
The first-generation sporadic simple groups encompass a collection of six finite simple groups discovered between 1967 and 1973, distinct from the Mathieu, Janko, Conway, and Fischer families. These groups were identified through computational searches and characterizations based on centralizers of involutions or automorphism groups of combinatorial structures, contributing to the early phases of the classification of finite simple groups. Their orders range from approximately 4 × 10^7 to 4 × 10^11, and they exhibit rank-3 permutation representations, often linked to graphs or designs. Unlike later sporadics, these groups were constructed without direct ties to the Leech lattice or the Monster group, relying instead on graph-theoretic or geometric methods prevalent in the 1960s and 1970s.[43] The Higman–Sims group HS, discovered by Donald G. Higman and Charles C. Sims in 1968, has order 44,352,000 = 2^9 · 3^2 · 5^3 · 7 · 11 and acts as a rank-3 permutation group on 100 points as the automorphism group of the Higman–Sims graph, a strongly regular graph with parameters (100, 22, 0, 6). This construction embeds HS within the full automorphism group of the graph, which has index 2 over HS, and highlights its 2-transitive action on certain subsets. HS contains a subgroup isomorphic to U_4(3):4, underscoring its connections to groups of Lie type, though it remains sporadic. The group's simplicity was established through character theory and subgroup analysis in its original presentation.[44] The McLaughlin group McL, identified by Jack McLaughlin in 1969, possesses order 898,128,000 = 2^7 · 3^6 · 5^3 · 7 · 11 and arises as a rank-3 automorphism group of the McLaughlin graph on 275 vertices, a strongly regular graph with parameters (275, 112, 30, 56). This graph is derived from the Steiner system S(5,8,24), the Witt design, providing a geometric construction that links McL to extremal combinatorial structures. McL features involution centralizers of type 2^{1+6}_+ · O_7(3), and its 5-rank (number of Sylow 5-subgroups) is 66, reflecting its intricate subgroup lattice. The group's uniqueness follows from characterizations of its 2-local structure. Dieter Held constructed the Held group He in 1969, a sporadic simple group of order 4,030,387,200 = 2^{10} · 3^3 · 5^2 · 7^3 · 17 acting 3-transitively on 51 points in its minimal degree representation. Its construction stems from the centralizer of an involution of type 2A, leading to a characterization via a (4,5)-arc in projective geometry or the Horton-Held graph on 51 vertices. He embeds a subgroup isomorphic to L_3(4):2^2, and its outer automorphism group is trivial. The simplicity proof relies on the analysis of its maximal subgroups, including U_3(5):4 and PSL(2,49). The Rudvalis group Ru, announced by Arunas Rudvalis in 1973, has order 145,926,144,000 = 2^{14} · 3^3 · 5^3 · 7 · 13 and acts primitively on 4060 points as an extension of a rank-3 group derived from the centralizer of an involution with structure 2^{1+8}_- · Sp_6(2). Its construction involves a permutation representation on cosets of a maximal subgroup of index 4060, linked to a graph on 28,680 vertices. Ru contains subgroups like U_5(2):2 and McL:2, and is 3-transitive on 28 points in another representation. Uniqueness is proven through exhaustive subgroup classification. Michio Suzuki discovered the Suzuki group Suz in 1968, with order 448,345,497,600 = 2^{13} · 3^7 · 5^2 · 7 · 11 · 13, constructed as an index-2 subgroup of the automorphism group of the Suzuki graph, a rank-3 graph on 1782 vertices with parameters (1782, 416, 130, 112). This graph arises from extensions of Ree groups of type ^2B_2(8), providing a bridge to Lie-type structures, though Suz itself is sporadic. It acts 3-transitively on 12 points and has involution centralizers of type 2^{1+6}_+ · O_6^-(2), with a Schur multiplier of order 3. The group's properties were verified using modular character tables. Finally, the O'Nan group O'N, found by Michael O'Nan in 1973 and fully characterized by 1976, is a sporadic simple group of order 460,815,505,920 = 2^9 · 3^4 · 5 · 7^3 · 11 · 19 · 31, acting as a rank-3 permutation group on 30,927 points via cosets of a point stabilizer. Its construction originates from a characterization of groups with a specific Sylow 2-subgroup of order 2^9 and an involution centralizer involving 2^4 : (3 × A_6). O'N has a double cover and outer automorphism of order 2, and contains maximal subgroups like 3^4 : (2^4 · A_8). Simplicity follows from the uniqueness of its 2-local geometry.| Group | Order | Discovery Year | Key Construction |
|---|---|---|---|
| HS | 44,352,000 | 1968 | Automorphism group of Higman–Sims graph (100 vertices) |
| McL | 898,128,000 | 1969 | Automorphism group of McLaughlin graph (275 vertices) |
| He | 4,030,387,200 | 1969 | Centralizer of involution in rank-3 action on 51 points |
| Ru | 145,926,144,000 | 1973 | Index-4060 permutation representation from involution centralizer |
| Suz | 448,345,497,600 | 1968 | Index-2 in automorphism group of Suzuki graph (1782 vertices) |
| O'N | 460,815,505,920 | 1973 | Characterization by Sylow 2-subgroup and rank-3 action on 30,927 points |
Second-generation sporadics
The second-generation sporadic simple groups—Harada–Norton (HN), Lyons (Ly), Thompson (Th), and Baby Monster (B)—represent a transitional class in the classification of finite simple groups, discovered during the 1970s and bridging earlier sporadics with the largest known sporadic, the Monster. These groups exhibit complex p-local structures for small primes like 2, 3, and 5, which facilitated their identification through computational and geometric methods. Unlike the first-generation sporadics, they lack small-degree transitive representations but feature intricate subgroup lattices and connections to higher-dimensional geometries, such as lattices in exceptional Lie algebras. Their discovery completed key gaps in the CFSG during the 1970s push toward full classification. The Harada–Norton group HN has order $2^{14} \cdot 3^6 \cdot 5^6 \cdot 7 \cdot 11 \cdot 19 = 273{,}030{,}912{,}000{,}000. It was identified independently by Y. Harada and S. P. Norton around 1975–1976 through analysis of potential simple groups with specific centralizer structures. HN is constructed as a permutation group of degree 1,140,000, arising from computations on its 5-local geometry, and admits a faithful representation of minimal degree 133 over the complex numbers. Its Schur multiplier is trivial, and the outer automorphism group has order 2; notable p-local features include a maximal 2-local subgroup isomorphic to $2^{1+12}_+ \cdot O_6^-(2) and a 3-local structure involving extensions of unitary groups. These properties underscore HN's role in probing large sporadic structures.[45][46] The Lyons group Ly has order $2^8 \cdot 3^7 \cdot 5^6 \cdot 7 \cdot 11 \cdot 31 \cdot 37 \cdot 67 = 51{,}765{,}179{,}004{,}000{,}000. Discovered by R. Lyons in 1972 via examination of 5-local geometries, it was proven simple and unique in subsequent work using Brauer characters and subgroup analyses. Ly is constructed through its action on a 111-dimensional module over \mathbb{F}_5, with a maximal 5-local subgroup $5^{1+6}_+ \cdot (3 \times \mathrm{Sp}_6(2)) \cdot 2. It possesses a trivial Schur multiplier and no outer automorphisms, featuring large 2- and 3-local subgroups like $2^8 : (\mathrm{SL}_3(3) \times 3) that highlight its pariah status among sporadics. Computational verifications confirmed its irreducibility in low-characteristic representations.[47][48] The Thompson group Th has order $2^{15} \cdot 3^{10} \cdot 5^3 \cdot 7^2 \cdot 13 \cdot 31 \cdot 37 \cdot 67 = 90{,}745{,}943{,}887{,}872{,}000. J. G. Thompson identified it in 1973 as a simple group emerging from studies of centralizers in potential larger sporadics. Th is constructed as the automorphism group of an even lattice in the 248-dimensional Lie algebra of E_8, preserving the lattice modulo 3 but not the full Lie bracket; this embedding into E_8(\mathbb{F}_3) provides a geometric realization.[49] With a trivial Schur multiplier and outer automorphism group of order 2, it exhibits rich p-local geometry, including a maximal 3-local subgroup $3^{1+10}_+ \cdot (3^5 : 2^4) and connections to Ree groups of type ^2G_2. Its uniqueness was established through exhaustive character table computations. The Baby Monster group B has order $2^{41} \cdot 3^{13} \cdot 5^7 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71 = 4{,}154{,}781{,}481{,}226{,}426{,}191{,}177{,}580{,}544{,}000{,}000{,}000. Predicted by B. Fischer in 1973 and constructed by R. L. Griess in 1980 as the automorphism group of a 196,883-dimensional algebra over \mathbb{F}_2, it was verified simple through detailed subgroup analysis. B has Schur multiplier of order 2 and trivial outer automorphism group, with prominent p-local structures such as a maximal 2-local subgroup of type $2^{1+24}_+ \cdot \mathrm{Co}_1 and 47-local geometry involving extensions of PSL structures. Its vast size and modular representations, including a minimal faithful degree of 9,725, link it to moonshine phenomena without direct Monster overlap here.[50]Monster and related groups
The Monster group M, also known as the Fischer–Griess Monster, is the largest of the 26 sporadic finite simple groups.[51] Its order is |M| = 2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71, which is approximately $8 \times 10^{53}.[51] The existence of this group was first predicted in 1973 independently by Bernd Fischer and Robert L. Griess, Jr., based on computational evidence from centralizer structures in related sporadic groups.[52] Griess provided the first explicit construction in 1982, realizing M as the automorphism group of a specific 196,884-dimensional commutative, non-associative algebra over the real numbers, known as the Griess algebra, which carries an invariant positive definite symmetric bilinear form.[52] This algebra has a 1-dimensional fixed subspace under the group action, with M acting faithfully and irreducibly on the orthogonal complement, a 196,883-dimensional representation that serves as the group's minimal faithful module.[52] An alternative construction was later given by John H. Conway in 1985, using the Leech lattice and related structures to generate M more accessibly via permutations and commutation relations.[53] The Monster's uniqueness as a simple group satisfying the structural conditions from Griess's work was established by John G. Thompson in the early 1980s, completing its identification within the classification of finite simple groups during that decade.[52] In 1984, Igor B. Frenkel, James Lepowsky, and Arne Meurman constructed the moonshine module, a vertex operator algebra V^\natural of central charge 24 on which M acts naturally, providing a graded representation whose character is the j-function from modular forms and linking the group to deeper connections in number theory and physics. Centralizers of certain elements in M yield other large sporadic groups as quotients or subgroups, including the Baby Monster group B (the centralizer of an involution) and the Harada–Norton group HN, highlighting the Monster's role as a central hub among the sporadics.[52] These relations underscore M's exceptional size and interconnectedness, distinguishing it as the capstone of the sporadic families.[51]Simple groups of small order
Non-cyclic examples up to 10^6
The non-cyclic finite simple groups of order less than 1,000,000 number 56 in total and fall into three categories: alternating groups A_n for $5 \leq n \leq 9, groups of Lie type (primarily low-rank projective special linear groups PSL(d,q), along with a few others like the projective special unitary group PSU(3,3)), and four sporadic groups (the Mathieu groups M_{11} and M_{12}, the first two Janko groups J_1 and J_2).[54] These groups were fully enumerated by Marshall Hall in 1972, prior to the complete classification of finite simple groups, with no additional simple groups of this order range discovered since.[54] The classification confirms that no non-abelian simple groups exist between orders 1 and 59, making A_5 the smallest such group.[54] The alternating groups in this range provide foundational examples of simple groups arising from symmetries of sets, while the Lie-type groups, particularly the PSL(2,q) family (where q is a prime power), dominate the list due to their prevalence for small q. Isomorphisms between some of these groups highlight deep structural connections; for instance, A_5 \cong PSL(2,5), A_6 \cong PSL(2,9), and PSL(2,7) \cong PSL(3,2). The sporadic groups appear only at the smaller end of this order range, with M_{11} being the smallest at order 7,920. The following table lists selected non-cyclic simple groups of order less than 10,000, emphasizing alternating and small Lie-type examples, with noted isomorphisms where applicable. Orders are computed via standard formulas, such as |A_n| = n!/2 and |PSL(2,q)| = q(q^2-1)/\gcd(2,q-1).[54]| Order | Group | Isomorphism Notes |
|---|---|---|
| 60 | A_5 | PSL(2,5) \cong PSp(2,5) |
| 168 | PSL(2,7) | PSL(3,2) \cong PSp(2,7) |
| 360 | A_6 | PSL(2,9) \cong PSp(2,9) |
| 504 | PSL(2,8) | \cong ^2G_2(3)' (Ree group) |
| 660 | PSL(2,11) | |
| 1,092 | PSL(2,13) | |
| 2,448 | PSL(2,17) | |
| 2,520 | A_7 | |
| 3,420 | PSL(2,19) | |
| 4,080 | PSL(2,16) | |
| 5,616 | PSL(3,3) | |
| 6,048 | PSU(3,3) | |
| 6,072 | PSL(2,23) | |
| 7,800 | PSL(2,25) | |
| 7,920 | M_{11} | (sporadic) |
Orders and classifications
The classification of finite simple groups by order provides key insights into their structure and distribution, particularly for small orders where explicit enumeration is feasible. The smallest order of a non-abelian finite simple group is 60, realized by the alternating group A_5, which is isomorphic to \mathrm{PSL}_2(5).[55] There are no non-abelian simple groups of order less than 60, as demonstrated by applying Sylow theorems to show that any candidate group would possess a normal Sylow subgroup, contradicting simplicity.[56] Up to order $10^3, four non-abelian simple groups exist: A_5 (60), \mathrm{PSL}_2(7) (168), A_6 (360), and \mathrm{PSL}_2(11) (660).[55] Classifications for small orders rely on foundational tools such as Sylow theorems, which constrain possible group orders by requiring the number of Sylow p-subgroups to be congruent to 1 modulo p and greater than 1 for all primes p (to avoid normal subgroups), thereby excluding many candidate orders.[57] Representation theory complements these bounds by analyzing character degrees and irreducible representations to identify groups as known families, such as alternating or linear groups, often via embedding into matrix groups over finite fields.[58] For instance, Burnside's p q-theorem rules out non-abelian simple groups of order p^a q^b except for specific cases like A_5.[57] A pivotal historical contribution was Leonard Eugene Dickson's classification of the projective special linear groups \mathrm{PSL}_2(q) over finite fields, completed in his 1901 monograph Linear Groups with an Exposition of the Galois Field Theory, which enumerated these groups and their isomorphisms for small q.[59] This work identified many of the smallest non-abelian simple groups, such as \mathrm{PSL}_2(7) and \mathrm{PSL}_2(11).[59] All non-abelian finite simple groups of order less than $10^6 have been classified, comprising alternating groups A_n for $5 \leq n \leq 9, groups of Lie type (including various \mathrm{PSL}_d(q), \mathrm{PSU}_d(q), and others), and four sporadic groups: the Mathieu groups M_{11} (order 7920) and M_{12} (order 95040), along with Janko groups J_1 (order 175560) and J_2 (order 604800).[60] No additional sporadics appear below this threshold, and the classification confirms no undiscovered families in this range.[60] The following table summarizes the distribution of non-abelian simple groups by order up to $10^4, highlighting the rapid increase in examples from Lie-type families:| Order Range | Number | Representative Examples |
|---|---|---|
| 1–59 | 0 | None |
| 60–999 | 4 | A_5 (60), \mathrm{PSL}_2(7) (168), A_6 (360), \mathrm{PSL}_2(11) (660) |
| 1,000–9,999 | 13 | \mathrm{PSL}_2(13) (1,092), A_7 (2,520), \mathrm{PSL}_2(17) (2,448), M_{11} (7,920) |