Classification of finite simple groups
Encyclopedia
In mathematics
, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups
can be seen as the basic building blocks of all finite group
s, in much the same way as the prime number
s are the basic building blocks of the natural number
s. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups.
The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Gorenstein
, Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.
The classification theorem has applications in many branches of mathematics, as questions about the structure of finite group
s (and their action on other mathematical objects) can sometimes be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can sometimes be answered by checking each family of simple groups and each sporadic group.
Daniel Gorenstein
announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin group
s. The completed proof of the classification was announced by after Aschbacher and Smith published a 1221 page proof for the missing quasithin case.
wrote a 3rd volume covering the remaining characteristic 2 case. The proof can be broken up into several major pieces as follows:
The simple groups of small 2-rank include:
The classification of groups of small 2-rank, especially ranks at most 2, makes heavy use of ordinary and modular character theory, which is almost never directly used elsewhere in the classification.
All groups not of small 2 rank can be split into two major classes: groups of component type and groups of characteristic 2 type. This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem
implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type. (For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor
theorem only work for groups with elementary abelian subgroups of rank at least 3.)
These are more or less the groups of Lie type of odd characteristic of large rank, and alternating groups, together with some sporadic groups.
A major step in this case is to eliminate the obstruction of the core of an involution. This is accomplished by the B-theorem
, which states that every component of C/O(C) is the image of a component of C.
The idea is that these groups have a centralizer of an involution with a component that is a smaller quasisimple group, which can be assumed to be already known by induction. So to classify these groups one takes every central extension of every known finite simple group, and finds all simple groups with a centralizer of involution with this as a component. This gives a rather large number of different cases to check: there are not only 26 sporadic groups and 16 families of groups of Lie type and the alternating groups, but also many of the groups of small rank or over small fields behave differently from the general case and have to be treated separately, and the groups of Lie type of even and odd characteristic are also quite different.
As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, plus a handful of others that are alternating or sporadic or of odd characteristic. Their classification is divided into the small and large rank cases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which is often (but not always) the same as the rank of a Cartan subalgebra when the group is a group of Lie type in characteristic 2.
The rank 1 groups are the thin group
s, classified by Aschbacher, and the rank 2 ones are the notorious quasithin group
s, classified by Aschbacher and Smith. These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2.
Groups of rank at least 3 are further subdivided into 3 classes by the trichotomy theorem
, proved by Aschbacher for rank 3 and by Gorenstein and Lyons for rank at least 4.
The three classes are groups of GF(2) type (classified mainly by Timmesfeld), groups of "standard type" for some odd prime (classified by the Gilman-Griess theorem and work by several others), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups.
The general higher rank case consists mostly of the groups of Lie type over fields of characteristic 2 of rank at least 3 or 4.
.
As of 2005, six volumes of the second generation proof have been published , with most of the balance of the proof in manuscript. It is estimated that the new proof will eventually fill approximately 5,000 pages. (This length stems in part from second generation proof being written in a more relaxed style.) Aschbacher and Smith wrote their two volumes devoted to the quasithin case in such a way that those volumes can be part of the second generation proof.
Gorenstein and his collaborators have given several reasons why a simpler proof is possible.
has called the work on the classification problem by Ulrich Meierfrankenfeld, Bernd Stellmacher, Gernot Stroth, and a few others, a third generation program. One goal of this is to treat all groups in characteristic 2 uniformly using the amalgam method.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
can be seen as the basic building blocks of all finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s, in much the same way as the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s are the basic building blocks of the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups.
The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Gorenstein
Daniel Gorenstein
Daniel E. Gorenstein was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertation Gorenstein rings...
, Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.
Statement of the classification theorem
Theorem. Every finite simple group is isomorphic to one of the following groups:- A cyclic groupCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
with prime order; - An alternating group of degree at least 5;
- A simple group of Lie typeGroup of Lie typeIn mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
, including both- the classical Lie groupsClassical groupIn mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...
, namely the groups of projective special linear, unitaryUnitary groupIn mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
, symplecticSymplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
, or orthogonalOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
transformations over a finite fieldFinite fieldIn abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
; - the exceptional and twisted groups of Lie type (including the Tits groupTits groupIn mathematics, the Tits group 2F4′ is a finite simple group of order 17971200 = 211 · 33 · 52 · 13 found by ....
which is not strictly a group of Lie typeGroup of Lie typeIn mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
).
- the classical Lie groups
- The 26 sporadic simple groupSporadic groupIn the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
s.
The classification theorem has applications in many branches of mathematics, as questions about the structure of finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s (and their action on other mathematical objects) can sometimes be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can sometimes be answered by checking each family of simple groups and each sporadic group.
Daniel Gorenstein
Daniel Gorenstein
Daniel E. Gorenstein was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertation Gorenstein rings...
announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin group
Quasithin group
In mathematics, a quasithin group is roughly a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an...
s. The completed proof of the classification was announced by after Aschbacher and Smith published a 1221 page proof for the missing quasithin case.
Overview of the proof of the classification theorem
wrote two volumes outlining the low rank and odd characteristic part of the proof, andwrote a 3rd volume covering the remaining characteristic 2 case. The proof can be broken up into several major pieces as follows:
Groups of small 2-rank
The simple groups of low 2-rank are mostly groups of Lie type of small rank over fields of odd characteristic, together with five alternating and seven characteristic 2 type and nine sporadic groups.The simple groups of small 2-rank include:
- Groups of 2-rank 0, in other words groups of odd order, which are all solvable by the Feit-Thompson theorem.
- Groups of 2-rank 1. The Sylow 2-subgroups are either cyclic, which is easy to handle using the transfer map, or generalized quaternionQuaternionIn mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
, which are handled with the Brauer-Suzuki theorem: in particular there are no simple groups of 2-rank 1. - Groups of 2-rank 2. Alperin showed that the Sylow subgoup must be dihedral, quasidihedral, wreathed, or a Sylow 2-subgroup of U3(4). The first case was done by the Gorenstein–Walter theoremGorenstein–Walter theoremIn mathematics, the Gorenstein–Walter theorem, proved by , states that if a finite group G has a dihedral Sylow 2-subgroup, and O is the maximal normal subgroup of odd order, then G/O is isomorphic to a 2-group, or the alternating group A7, or a subgroup of PΓL22 containing PSL2 for q an odd prime...
which showed that the only simple groups are isomorphic to L2(q) for q odd or A7, the second and third cases were done by the Alperin–Brauer–Gorenstein theorem which implies that the only simple groups are isomorphic to L3(q) or U3(q) for q odd or M11, and the last case was done by Lyons who showed that U3(4) is the only simple possibility. - Groups of sectional 2-rank at most 4, classified by the Gorenstein–Harada theoremGorenstein–Harada theoremIn mathematical finite group theory, the Gorenstein–Harada theorem, proved by in a 464 page paper, classifies the simple finite groups of sectional 2-rank at most 4...
.
The classification of groups of small 2-rank, especially ranks at most 2, makes heavy use of ordinary and modular character theory, which is almost never directly used elsewhere in the classification.
All groups not of small 2 rank can be split into two major classes: groups of component type and groups of characteristic 2 type. This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem
Balance theorem
In mathematical finite group theory, the balance theorem states that if G is a group with no core then G either has disconnected Sylow 2-subgroups or it is of characteristic 2 type or it is of component type ....
implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type. (For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor
Signalizer functor
In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem gives conditions under which a signalizer functor comes from a subgroup...
theorem only work for groups with elementary abelian subgroups of rank at least 3.)
Groups of component type
A group is said to be of component type if for some centralizer C of an involution, C/O(C) has a component (where O(C) is the core of C, the maximal normal subgroup of odd order).These are more or less the groups of Lie type of odd characteristic of large rank, and alternating groups, together with some sporadic groups.
A major step in this case is to eliminate the obstruction of the core of an involution. This is accomplished by the B-theorem
B-theorem
In mathematical finite group theory, the B-theorem states that if C is the centralizer of an involution of a finite group, then every component of C/O is the image of a component of C ....
, which states that every component of C/O(C) is the image of a component of C.
The idea is that these groups have a centralizer of an involution with a component that is a smaller quasisimple group, which can be assumed to be already known by induction. So to classify these groups one takes every central extension of every known finite simple group, and finds all simple groups with a centralizer of involution with this as a component. This gives a rather large number of different cases to check: there are not only 26 sporadic groups and 16 families of groups of Lie type and the alternating groups, but also many of the groups of small rank or over small fields behave differently from the general case and have to be treated separately, and the groups of Lie type of even and odd characteristic are also quite different.
Groups of characteristic 2 type
A group is of characteristic 2 type if the generalized Fitting subgroup F*(Y) of every 2-local subgroup Y is a 2-group.As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, plus a handful of others that are alternating or sporadic or of odd characteristic. Their classification is divided into the small and large rank cases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which is often (but not always) the same as the rank of a Cartan subalgebra when the group is a group of Lie type in characteristic 2.
The rank 1 groups are the thin group
Thin group
In mathematics, in the realm of group theory, a group is said to be thin if there is a finite upper bound on the girth of the Cayley graph induced by any finite generating set...
s, classified by Aschbacher, and the rank 2 ones are the notorious quasithin group
Quasithin group
In mathematics, a quasithin group is roughly a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an...
s, classified by Aschbacher and Smith. These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2.
Groups of rank at least 3 are further subdivided into 3 classes by the trichotomy theorem
Trichotomy theorem
In mathematical finite group theory, the trichotomy theorem divides the simple groups of characteristic 2 type and rank at least 3 into three classes...
, proved by Aschbacher for rank 3 and by Gorenstein and Lyons for rank at least 4.
The three classes are groups of GF(2) type (classified mainly by Timmesfeld), groups of "standard type" for some odd prime (classified by the Gilman-Griess theorem and work by several others), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups.
The general higher rank case consists mostly of the groups of Lie type over fields of characteristic 2 of rank at least 3 or 4.
Existence and uniqueness of the simple groups
The main part of the classification produces a characterization of each simple group. It is then necessary to check that there exists a simple group for each characterization and that it is unique. This gives a large number of separate problems; for example, the original proofs of existence and uniqueness of the monster totaled about 200 pages, and the identification of the Ree groups by Thompson and Bombieri was one of the hardest parts of the classification. Many of the existence proofs and some of the uniqueness proofs for the sporadic proofs originally used computer calculations, some of which have since been replaced by shorter hand proofs.Gorenstein's program
In 1972 announced a program for completing the classification of finite simple groups, consisting of the following 16 steps:- Groups of low 2-rank. This was essentially done by Gorenstein and Harada, who classified the groups with sectional 2-rank at most 4. Most of the cases of 2-rank at most 2 had been done by the time Gorenstein announced his program.
- The semisimplicity of 2-layers. The problem is to prove that the 2-layer of the centralizer of an involution in a simple group is semisimple.
- Standard form in odd characteristic. If a group has an involution with a 2-component that is a group of Lie type of odd characteristic, the goal is to show that it has a centralizer of involution in "standard form" meaning that a centralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of 2-rank 1.
- Classification of groups of odd type. The problem is to show that if a group has a centralizer of involution in "standard form" then it is a group of Lie type of odd characteristic. This was solved by Aschbacher's classical involution theoremClassical involution theoremIn mathematical finite group theory, the classical involution theorem of classifies simple groups with a classical involution and satisfying some other conditions, showing that they are mostly groups of Lie type over a field of odd characteristic...
. - Quasi-standard form
- Central involutions
- Classification of alternating groups. More precisely, show that if a simple group has
- Some sporadic groups
- Thin groups. The simple thin finite groups, those with 2-local p-rank at most 1 for odd primes p, were classified by Aschbacher in 1978
- Groups with a strongly p-embedded subgroup for p odd
- The signalizer functor method for odd primes. The main problem is to prove a signalizer functorSignalizer functorIn mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem gives conditions under which a signalizer functor comes from a subgroup...
theorem for nonsolvable signalizer functors. This was solved by McBride in 1982. - Groups of characteristic p type. This is the problem of groups with a strongly p-embedded 2-local subgroup with p odd, which was handled by Aschbacher.
- Quasithin groups. A quasithin groupQuasithin groupIn mathematics, a quasithin group is roughly a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an...
is one whose 2-local subgroups have p-rank at most 2 for all odd primes p, and the problem is to classify the simple ones of characteristic 2 type. This was completed by Aschbacher and Smith in 2004. - Groups of low 2-local 3-rank. This was essentially solved by Aschbacher's trichotomy theoremTrichotomy theoremIn mathematical finite group theory, the trichotomy theorem divides the simple groups of characteristic 2 type and rank at least 3 into three classes...
for groups with e(G)=3. The main change is that 2-local 3-rank is replaced by 2-local p-rank for odd primes. - Centralizers of 3-elements in standard form. This was essentially done by the Trichotomy theoremTrichotomy theoremIn mathematical finite group theory, the trichotomy theorem divides the simple groups of characteristic 2 type and rank at least 3 into three classes...
. - Classification of simple groups of characteristic 2 type. This was handled by the Gilman-Griess theorem, with 3-elements replaced by p-elements for odd primes.
Timeline of the proof
Many of the items in the list below are taken from . The date given is usually the publication date of the complete proof of a result, which is sometimes several years later than the proof or first announcement of the result, so some of the items appear in the "wrong" order.| Publication date | |
|---|---|
| 1832 | Galois introduces normal subgroups and finds the simple groups An (n≥5) and PSL2(Fp) (p≥5) |
| 1854 | Cayley defines abstract groups |
| 1861 | Mathieu finds the first two Mathieu group Mathieu group In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered... s M11, M12, the first sporadic simple groups. |
| 1870 | Jordan lists some simple groups: the alternating and projective special linear ones, and emphasizes the importance of the simple groups. |
| 1872 | Sylow proves the Sylow theorem Sylow theorem In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains... s |
| 1873 | Mathieu finds three more Mathieu group Mathieu group In the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered... s M22, M23, M24. |
| 1892 | Otto Hölder proves that the order of any nonabelian finite simple group must be a product of at least 4 primes, and asks for a classification of finite simple groups. |
| 1893 | Cole classifies simple groups of order up to 660 |
| 1896 | Frobenius and Burnside begin the study of character theory of finite groups. |
| 1899 | Burnside classifies the simple groups such that the centralizer of every involution is a non-trivial elementary abelian 2-group. |
| 1901 | Frobenius proves that a Frobenius group Frobenius group In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial elementfixes more than one point and some non-trivial element fixes a point.They are named after F. G. Frobenius.- Structure :... has a Frobenius kernel, so in particular is not simple. |
| 1901 | Dickson defines classical groups over arbitrary finite fields, and exceptional groups of type G2 over fields of odd characteristic. |
| 1901 | Dickson introduces the exceptional finite simple groups of type E6. |
| 1904 | Burnside uses character theory to prove Burnside's theorem that the order of any non-abelian finite simple group must be divisible by at least 3 distinct primes. |
| 1905 | Dickson introduces simple groups of type G2 over fields of even characteristic |
| 1911 | Burnside conjectures that every non-abelian finite simple group has even order |
| 1928 | Hall proves the existence of Hall subgroup Hall subgroup In mathematics, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They are named after the group theorist Philip Hall.- Definitions :A Hall divisor of an integer n is a divisor d of n such that... s of solvable groups |
| 1933 | Hall begins his study of p-groups |
| 1935 | Brauer begins the study of modular characters. |
| 1936 | Zassenhaus classifies finite sharply 3-transitive permutation groups |
| 1938 | Fitting introduces the Fitting subgroup Fitting subgroup In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it represents the smallest subgroup which "controls" the structure of G when G is solvable... and proves Fitting's theorem that for solvable groups the Fitting subgroup contains its centralizer. |
| 1942 | Brauer describes the modular characters of a group divisible by a prime to the first power. |
| 1954 | Brauer classifies simple groups with GL2(Fq) as the centralizer of an involution. |
| 1955 | The Brauer–Fowler theorem implies that the number of finite simple groups with given centralizer of involution is finite, suggesting an attack on the classification using centralizers of involutions. |
| 1955 | Chevalley introduces the Chevalley groups, in particular introducing exceptional simple groups of types F4, E7, and E8. |
| 1956 | Hall–Higman theorem Hall–Higman theorem In mathematical group theory, the Hall–Higman theorem, due to , describes the possibilities for the minimal polynomial of an element of prime power order for a representation of a p-solvable group.-Statement:... |
| 1957 | Suzuki shows that all finite simple CA group CA group In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in... s of odd order are cyclic. |
| 1958 | The Brauer–Suzuki–Wall theorem Brauer–Suzuki–Wall theorem In mathematics, the Brauer–Suzuki–Wall theorem, proved by , characterizes the one-dimensional unimodular projective groups over finite fields.-References:... characterizes the projective special linear groups of rank 1, and classifies the simple CA group CA group In mathematics, in the realm of group theory, a group is said to be a CA-group or centralizer abelian group if the centralizer of any nonidentity element is an abelian subgroup. Finite CA-groups are of historical importance as an early example of the type of classifications that would be used in... s. |
| 1959 | Steinberg introduces the Steinberg groups, giving some new finite simple groups, or types 3D4 and 2E6 (the latter were independently found at about the same time by Jacques Tits). |
| 1959 | The Brauer–Suzuki theorem Brauer–Suzuki theorem In mathematics, the Brauer–Suzuki theorem, proved by , , , states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group has a centre of order 2. In particular, such a group cannot be simple.A generalization of the... about groups with generalized quaternion Sylow 2-subgroups shows in particular that none of them are simple. |
| 1960 | Thompson proves that a group with a fixed-point-free automorphism of prime order is nilpotent. |
| 1960 | Feit, Hall, and Thompson show that all finite simple CN group CN group In mathematics, in the area of algebra known as group theory, a more than fifty-year effort was made to answer a conjecture of : are all groups of odd order solvable? Progress was made by showing that CA-groups, groups in which the centralizer of a non-identity element is abelian, of odd order are... s of odd order are cyclic. |
| 1960 | Suzuki introduces the Suzuki groups, with types 2B2. |
| 1961 | Ree introduces the Ree groups, with types 2F4 and 2G2. |
| 1963 | Feit and Thompson prove the odd order theorem. |
| 1964 | Tits introduces BN pairs for groups of Lie type and finds the Tits group Tits group In mathematics, the Tits group 2F4′ is a finite simple group of order 17971200 = 211 · 33 · 52 · 13 found by .... |
| 1965 | The Gorenstein–Walter theorem Gorenstein–Walter theorem In mathematics, the Gorenstein–Walter theorem, proved by , states that if a finite group G has a dihedral Sylow 2-subgroup, and O is the maximal normal subgroup of odd order, then G/O is isomorphic to a 2-group, or the alternating group A7, or a subgroup of PΓL22 containing PSL2 for q an odd prime... classifies groups with a dihedral Sylow 2-subgroup. |
| 1966 | Glauberman proves the Z* theorem |
| 1966 | Janko introduces the Janko group J1 Janko group J1 In mathematics, the smallest Janko group, J1, is a simple sporadic group of order 175560. It was originally described by Zvonimir Janko and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century... , the first new sporadic group for about a century. |
| 1968 | Glauberman proves the ZJ theorem ZJ theorem In mathematics, George Glauberman's ZJ theorem states that if a finite group G is p-constrained and p-stable and has a normal p-subgroup for some odd prime p, then Op′Z is a normal subgroup of G, for any Sylow p-subgroup S.-Notation and definitions:*J is the Thompson subgroup of a p-group S:... |
| 1968 | Higman and Sims introduce the Higman–Sims group |
| 1968 | Conway introduces the Conway groups |
| 1969 | Walter's theorem classifies groups with abelian Sylow 2-subgroups |
| 1969 | Introduction of the Suzuki sporadic group Suzuki sporadic group In mathematical group theory, the Suzuki group Suz or Sz is a sporadic simple group of order 213 · 37 · 52· 7 · 11 · 13 = 448345497600 discovered by . It is not related to the Suzuki groups of Lie type.... , the Janko group J2, the Janko group J3 Janko group J3 In mathematics, the third Janko group J3, also known as the Higman-Janko-McKay group, is a finite simple sporadic group of order 50232960. Evidence for its existence was uncovered by , and it was shown to exist by... , the McLaughlin group McLaughlin group (mathematics) In the mathematical discipline known as group theory, the McLaughlin group McL is a sporadic simple group of order 27 · 36 · 53· 7 · 11 = 898,128,000, discovered by as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with 275 =1+112+162 vertices... , and the Held group Held group In the mathematical field of group theory, the Held group He is one of the 26 sporadic simple groups, and has order... . |
| 1969 | Gorenstein introduces signalizer functor Signalizer functor In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem gives conditions under which a signalizer functor comes from a subgroup... s based on Thompson's ideas. |
| 1970 | Bender introduced the generalized Fitting subgroup |
| 1970 | The Alperin–Brauer–Gorenstein theorem classifies groups with quasi-dihedral or wreathed Sylow 2-subgroups, completing the classification of the simple groups of 2-rank at most 2 |
| 1971 | Fischer introduces the three Fischer groups |
| 1971 | Thompson classifies quadratic pair Quadratic pair In mathematical finite group theory, a quadratic pair for the odd prime p, introduced by , is a finite group G together with a quadratic module, a faithful representation M on a vector space over the finite field with p elements such that G is generated by elements with minimum polynomial 2... s |
| 1971 | Bender classifies group with a strongly embedded subgroup Strongly embedded subgroup In finite group theory, an area of abstract algebra, a strongly embedded subgroup of a finite group G is a proper subgroup H of even order such that H ∩ Hg has odd order whenever g is not in H.... |
| 1972 | Gorenstein proposes a 16-step program for classifying finite simple groups; the final classification follows his outline quite closely. |
| 1972 | Lyons introduces the Lyons group Lyons group In the mathematical field of group theory, the Lyons group Ly , is a sporadic simple group of order... |
| 1973 | Rudvalis introduces the Rudvalis group Rudvalis group In the mathematical field of group theory, the Rudvalis group Ru is a sporadic simple group of order-Properties:The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer the Ree group... |
| 1973 | Fisher discovers the baby monster group Baby Monster group In the mathematical field of group theory, the Baby Monster group B is a group of orderThe Baby Monster group is one of the sporadic groups, and has the second highest order of these, with the highest order being that of the Monster group... (unpublished), which Fischer and Griess use to discover the monster group Monster group In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:... , which in turn leads Thompson to the Thompson sporadic group and Norton to the Harada–Norton group (also found in a different way by Harada). |
| 1974 | Thompson classifies N-groups N-group (finite group theory) In mathematical finite group theory, an N-group is a group all of whose local subgroups are solvable groups... , groups all of whose local subgroups are solvable. |
| 1974 | The Gorenstein–Harada theorem Gorenstein–Harada theorem In mathematical finite group theory, the Gorenstein–Harada theorem, proved by in a 464 page paper, classifies the simple finite groups of sectional 2-rank at most 4... classifies the simple groups of sectional 2-rank at most 4, dividing the remaining finite simple groups into those of component type and those of characteristic 2 type. |
| 1974 | Tits shows that groups with BN pairs of rank at least 3 are groups of Lie type |
| 1974 | Aschbacher classifies the groups with a proper 2-generated core |
| 1975 | Gorenstein and Walter prove the L-balance theorem L-balance theorem In mathematical finite group theory, the L-balance theorem was proved by .The letter L stands for the layer of a group, and "balance" refers to the property discussed below.-Statement:... |
| 1976 | Glauberman proves the solvable signalizer functor Signalizer functor In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem gives conditions under which a signalizer functor comes from a subgroup... theorem |
| 1976 | Aschbacher proves the component theorem Component theorem In the mathematical classification of finite simple groups, the component theorem of shows that if G is a simple group of odd type, and various other assumptions are satisfied, then G has a centralizer of an involution with a "standard component" with small centralizer.... , showing roughly that groups of odd type satisfying some conditions have a component in standard form. The groups with a component of standard form were classified in a large collection of papers by many authors. |
| 1976 | O'Nan introduces the O'Nan group O'Nan group In the mathematical field of group theory, the O'Nan group O'N is a sporadic simple group of orderThe Schur multiplier has order 3, and its outer automorphism group has order 2... |
| 1976 | Janko introduces the Janko group J4 Janko group J4 In mathematics, the fourth Janko group J4 is the sporadic finite simple group of order 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 = 86775571046077562880 whose existence was suggested by Zvonimir Janko . Its existence and uniqueness was shown by Simon P. Norton and others in 1980... , the last sporadic group to be discovered |
| 1977 | Aschbacher characterizes the groups of Lie type of odd characteristic in his classical involution theorem Classical involution theorem In mathematical finite group theory, the classical involution theorem of classifies simple groups with a classical involution and satisfying some other conditions, showing that they are mostly groups of Lie type over a field of odd characteristic... . After this theorem, which in some sense deals with "most" of the simple groups, it was generally felt that the end of the classification was in sight. |
| 1978 | Timmesfeld proves the O2 extraspecial theorem, breaking the classification of groups of GF(2)-type into several smaller problems. |
| 1978 | Aschbacher classifies the thin finite group Thin finite group In the mathematical classification of finite groups, a thin group is a finite group such that for any odd prime the Sylow p-subgroups of the 2-local subgroups are cyclic.... s, which are mostly rank 1 groups of Lie type over fields of even characteristic. |
| 1981 | Bombieri uses elimination theory to complete Thompson's work on the characterization of Ree groups, one of the hardest steps of the classification. |
| 1982 | McBride proves the signalizer functor theorem for all finite groups. |
| 1982 | Griess constructs the monster group Monster group In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:... by hand |
| 1983 | The Gilman–Griess theorem Gilman–Griess theorem In finite group theory, a mathematical discipline, the Gilman–Griess theorem, proved by , classifies the finite simple groups of characteristic 2 type with e ≥ 4 that have a "standard component", which covers one of the three cases of the trichotomy theorem.... classifies groups groups of characteristic 2 type and rank at least 4 with standard components, one of the three cases of the trichotomy theorem. |
| 1983 | Aschbacher proves that no finite group satisfies the hypothesis of the uniqueness case Uniqueness case In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem.... , one of the three cases given by the trichotomy theorem for groups of characteristic 2 type. |
| 1983 | Gorenstein and Lyons prove the trichotomy theorem Trichotomy theorem In mathematical finite group theory, the trichotomy theorem divides the simple groups of characteristic 2 type and rank at least 3 into three classes... for groups of characteristic 2 type and rank at least 4, while Aschbacher does the case of rank 3. This divides these groups into 3 subcases: the uniqueness case, groups of GF(2) type, and groups with a standard component. |
| 1983 | Gorenstein announces the proof of the classification is complete, somewhat prematurely as the proof of the quasithin case was incomplete. |
| 1994 | Gorenstein, Lyons, and Solomon begin publication of the revised classification |
| 2004 | Aschbacher and Smith publish their work on quasithin group Quasithin group In mathematics, a quasithin group is roughly a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an... s (which are mostly groups of Lie type of rank at most 2 over fields of even characteristic), filling the last (known) gap in the classification. |
Second-generation classification
The proof of the theorem, as it stood around 1985 or so, can be called first generation. Because of the extreme length of the first generation proof, much effort has been devoted to finding a simpler proof, called a second-generation classification proof. This effort, called "revisionism", was originally led by Daniel GorensteinDaniel Gorenstein
Daniel E. Gorenstein was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertation Gorenstein rings...
.
As of 2005, six volumes of the second generation proof have been published , with most of the balance of the proof in manuscript. It is estimated that the new proof will eventually fill approximately 5,000 pages. (This length stems in part from second generation proof being written in a more relaxed style.) Aschbacher and Smith wrote their two volumes devoted to the quasithin case in such a way that those volumes can be part of the second generation proof.
Gorenstein and his collaborators have given several reasons why a simpler proof is possible.
- The most important is that the correct, final statement of the theorem is now known. Simpler techniques can be applied that are known to be adequate for the types of groups we know to be finite simple. In contrast, those who worked on the first generation proof did not know how many sporadic groups there were, and in fact some of the sporadic groups (e.g., the Janko groupJanko groupIn mathematics, a Janko group is one of the four sporadic simple groups named for Zvonimir Janko. Janko constructed the first Janko group J1 in 1965. At the same time, Janko also predicted the existence of J2 and J3. In 1976, he suggested the existence of J4...
s) were discovered while proving other cases of the classification theorem. As a result, many of the pieces of the theorem were proved using techniques that were overly general.
- Because the conclusion was unknown, the first generation proof consists of many stand-alone theorems, dealing with important special cases. Much of the work of proving these theorems was devoted to the analysis of numerous special cases. Given a larger, orchestrated proof, dealing with many of these special cases can be postponed until the most powerful assumptions can be applied. The price paid under this revised strategy is that these first generation theorems no longer have comparatively short proofs, but instead rely on the complete classification.
- Many first generation theorems overlap, and so divide the possible cases in inefficient ways. As a result, families and subfamiles of finite simple groups were identified multiple times. The revised proof eliminates these redundancies by relying on a different subdivision of cases.
- Finite group theorists have more experience at this sort of exercise, and have new techniques at their disposal.
has called the work on the classification problem by Ulrich Meierfrankenfeld, Bernd Stellmacher, Gernot Stroth, and a few others, a third generation program. One goal of this is to treat all groups in characteristic 2 uniformly using the amalgam method.
External links
- ATLAS of Finite Group Representations. Searchable database of representationsGroup representationIn the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
and other data for many finite simple groups. - Elwes, Richard, "An enormous theorem: the classification of finite simple groups," Plus Magazine, Issue 41, December 2006. For laypeople.
- Madore, David (2003) Orders of nonabelian simple groups. Includes a list of all nonabelian simple groups up to order 1010.