Group of Lie type
Encyclopedia
In mathematics
, a group of Lie type G(k) is a (not necessarily finite) 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. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups.
and are standard references for groups of Lie type.
by . These groups were studied by L. E. Dickson and Jean Dieudonné
. Emil Artin
investigated the orders of such groups, with a view to classifying cases of coincidence.
A classical group is, roughly speaking, a special linear
, orthogonal
, symplectic
, or unitary group
. There are several minor variations of these, given by taking derived subgroups or central quotient
s, the latter yielding projective linear group
s. They can be constructed over finite fields (or any other field) in much the same way that they are constructed over the real numbers. They correspond to the series
An, Bn, Cn, Dn,
2An, 2Dn of Chevalley and Steinberg groups.
s, and the work of on Lie algebras, by means of which the Chevalley group concept was isolated. Chevalley constructed a Chevalley basis
(a sort of integral form) for all the complex simple Lie algebras (or rather of their universal enveloping algebra
s), which can be used to define the corresponding algebraic groups over the integers. In particular, he could take their points with values in any finite field. For the Lie algebras An, Bn, Cn, Dn this gave well known classical groups, but his construction also gave groups associated to the exceptional Lie algebras
E6, E7, E8, F4,
and G2. The ones of type G2 had already been constructed by , and the ones of type E6 by .
The unitary group arises as follows: the general linear group over the complex number
s has a diagram automorphism given by reversing the Dynkin diagram An (which corresponds to taking the transpose inverse), and a field automorphism given by taking complex conjugation, which commute. The unitary group is the group of fixed points of the product of these two automorphisms.
In the same way, many Chevalley groups have diagram automorphisms induced by automorphisms of their Dynkin diagrams, and field automorphisms induced by automorphisms of a finite field. Analogously to the unitary case, Steinberg constructed families of groups by taking fixed points of a product of a diagram and a field automorphism.
These gave:
The groups of type 3D4 have no analogue over the reals, as the complex numbers have no automorphism of order 3. The symmetries of the
diagram also give rise to triality
.
He found that if a finite field of characteristic 2 also has an automorphism whose square was the Frobenius map, then an analogue of Steinberg's construction gave the Suzuki groups. The fields with such an automorphism are those of order 22n+1, and the corresponding groups are the Suzuki groups
(Strictly speaking, the group Suz(2) is not counted as a Suzuki group as it is not simple: it is the Frobenius group
of order 20.) Ree was able to find two new similar families
and
of simple groups by using the fact that F4
and G2 have extra automorphisms in characteristic 2 and 3. (Roughly speaking, in characteristic p one is allowed to ignore the arrow on bonds of multiplicity p in the Dynkin diagram when taking diagram automorphisms.) The smallest group 2F4(2) of type 2F4 is not simple, but it has a simple subgroup of index
2, called the Tits group
(named after the mathematician Jacques Tits
). The smallest group 2G2(3) of type 2G2 is not simple, but it has a simple normal subgroup of index 3,
isomorphic to SL2(8). In the classification of finite simple groups
, the Ree groups
are the ones whose structure is hardest to pin down explicitly. These groups also played a role in the discovery of the first modern sporadic group. They have involution centralizers of the form Z/2Z × PSL2(q) for q = 3n, and by investigating
groups with an involution centralizer of the similar form Z/2Z × PSL2(5)
Janko found the sporadic group J1
.
The Suzuki groups are of great interest as the only finite non-abelian simple groups with order not divisible by 3. They have order (22(2n+1))(22(2n+1) + 1)(2(2n+1) − 1).
, symmetric
and alternating groups, with the projective special linear groups over prime finite fields, PSL(2,p) being constructed by Évariste Galois
in the 1830s. The systematic exploration of finite groups of Lie type started with Camille Jordan
's theorem that the projective special linear group PSL2(q) is simple for q ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSLn(q) of finite simple groups. Other classical groups were studied by Leonard Dickson in the beginning of 20th century. In the 1950s Claude Chevalley
realized that after an appropriate reformulation, many theorems about semisimple Lie groups admit analogues for algebraic groups over an arbitrary field k, leading to construction of what are now called Chevalley groups. Moreover, as in the case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups (Tits simplicity theorem). Although it was known since 19th century that other finite simple groups exist (for example, Mathieu groups), gradually a belief formed that nearly all finite simple groups can be accounted for by appropriate extensions of Chevalley's construction, together with cyclic and alternating groups. Moreover, the exceptions, the sporadic groups, share many properties with the finite groups of Lie type, and in particular, can be constructed and characterized based on their geometry in the sense of Tits.
The belief has now become a theorem – the classification of finite simple groups
. Inspection of the list of finite simple groups shows that groups of Lie type over a finite field
include all the finite simple groups other than the cyclic groups, the alternating groups, the Tits group
, and the 26 sporadic simple groups.
For a complete list of these exceptions see the list of finite simple groups.
Many of these special properties are related to certain sporadic simple groups.
Alternating groups sometimes behave as if they were groups of Lie type over the field with one element
.
Some of the small alternating groups also have exceptional properties.
The alternating groups usually have an outer automorphism group
of order 2, but the alternating group on 6 points has an outer automorphism group of order 4. Alternating groups usually have a Schur multiplier of order 2, but the ones on 6 or 7 points have a Schur multiplier of order 6.
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...
, a group of Lie type G(k) is a (not necessarily finite) group
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...
of rational points of a reductive linear algebraic group
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
G with values in the field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
k. Finite groups of Lie type form the bulk of nonabelian finite simple groups. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki–Ree groups.
and are standard references for groups of Lie type.
Classical groups
An initial approach to this question was the definition and detailed study of the so-called classical groups over finite and other fieldsField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
by . These groups were studied by L. E. Dickson and Jean Dieudonné
Jean Dieudonné
Jean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
. Emil Artin
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
investigated the orders of such groups, with a view to classifying cases of coincidence.
A classical group is, roughly speaking, a special linear
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
, orthogonal
Orthogonal group
In 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...
, symplectic
Symplectic group
In 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 unitary group
Unitary group
In 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...
. There are several minor variations of these, given by taking derived subgroups or central quotient
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
s, the latter yielding projective linear group
Projective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group is the induced action of the general linear group of a vector space V on the associated projective space P...
s. They can be constructed over finite fields (or any other field) in much the same way that they are constructed over the real numbers. They correspond to the series
An, Bn, Cn, Dn,
2An, 2Dn of Chevalley and Steinberg groups.
Chevalley groups
The theory was clarified by the theory of algebraic groupAlgebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s, and the work of on Lie algebras, by means of which the Chevalley group concept was isolated. Chevalley constructed a Chevalley basis
Chevalley basis
In mathematics, a Chevalley basis for a simple complex Lie algebra isa basis constructed by Claude Chevalley with the property that all structure constants are integers...
(a sort of integral form) for all the complex simple Lie algebras (or rather of their universal enveloping algebra
Universal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
s), which can be used to define the corresponding algebraic groups over the integers. In particular, he could take their points with values in any finite field. For the Lie algebras An, Bn, Cn, Dn this gave well known classical groups, but his construction also gave groups associated to the exceptional Lie algebras
E6, E7, E8, F4,
and G2. The ones of type G2 had already been constructed by , and the ones of type E6 by .
Steinberg groups
Chevalley's construction did not give all of the known classical groups: it omitted the unitary groups and the non-split orthogonal groups. found a modification of Chevalley's construction that gave these groups and two new families 3D4, 2E6, the second of which was discovered at about the same time from a different point of view by . This construction generalizes the usual construction of the unitary group from the general linear group.The unitary group arises as follows: the general linear group over the complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s has a diagram automorphism given by reversing the Dynkin diagram An (which corresponds to taking the transpose inverse), and a field automorphism given by taking complex conjugation, which commute. The unitary group is the group of fixed points of the product of these two automorphisms.
In the same way, many Chevalley groups have diagram automorphisms induced by automorphisms of their Dynkin diagrams, and field automorphisms induced by automorphisms of a finite field. Analogously to the unitary case, Steinberg constructed families of groups by taking fixed points of a product of a diagram and a field automorphism.
These gave:
- the unitary groups 2An, from the order 2 automorphism of An;
- further orthogonal groups 2Dn, from the order 2 automorphism of Dn;
- the new series 2E6, from the order 2 automorphism of E6;
- the new series 3D4, from the order 3 automorphism of D4.
The groups of type 3D4 have no analogue over the reals, as the complex numbers have no automorphism of order 3. The symmetries of the
diagram also give rise to trialityTriality
In mathematics, triality is a relationship between three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin, the double cover of 8-dimensional rotation group SO,...
.
Suzuki–Ree groups
found a new infinite series of groups that at first sight seemed unrelated to the known algebraic groups. knew that the algebraic group B2 had an "extra" automorphism in characteristic 2 whose square was the Frobenius automorphism.He found that if a finite field of characteristic 2 also has an automorphism whose square was the Frobenius map, then an analogue of Steinberg's construction gave the Suzuki groups. The fields with such an automorphism are those of order 22n+1, and the corresponding groups are the Suzuki groups
- 2B2(22n+1) = Suz(22n+1).
(Strictly speaking, the group Suz(2) is not counted as a Suzuki group as it is not simple: it is the 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 :...
of order 20.) Ree was able to find two new similar families
- 2F4(22n+1)
and
- 2G2(32n+1)
of simple groups by using the fact that F4
and G2 have extra automorphisms in characteristic 2 and 3. (Roughly speaking, in characteristic p one is allowed to ignore the arrow on bonds of multiplicity p in the Dynkin diagram when taking diagram automorphisms.) The smallest group 2F4(2) of type 2F4 is not simple, but it has a simple subgroup of index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
2, called 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 ....
(named after the mathematician Jacques Tits
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
). The smallest group 2G2(3) of type 2G2 is not simple, but it has a simple normal subgroup of index 3,
isomorphic to SL2(8). In the classification of finite simple groups
Classification of finite simple groups
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 groups, in much the same way as the prime numbers are the basic...
, the Ree groups
- 2G2(32n+1)
are the ones whose structure is hardest to pin down explicitly. These groups also played a role in the discovery of the first modern sporadic group. They have involution centralizers of the form Z/2Z × PSL2(q) for q = 3n, and by investigating
groups with an involution centralizer of the similar form Z/2Z × PSL2(5)
Janko found the sporadic group J1
Janko group
In 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...
.
The Suzuki groups are of great interest as the only finite non-abelian simple groups with order not divisible by 3. They have order (22(2n+1))(22(2n+1) + 1)(2(2n+1) − 1).
Relations with finite simple groups
Finite groups of Lie type were among the first groups to be considered in mathematics, after cyclicCyclic group
In 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...
, symmetric
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
and alternating groups, with the projective special linear groups over prime finite fields, PSL(2,p) being constructed by Évariste Galois
Évariste Galois
Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...
in the 1830s. The systematic exploration of finite groups of Lie type started with Camille Jordan
Camille Jordan
Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...
's theorem that the projective special linear group PSL2(q) is simple for q ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSLn(q) of finite simple groups. Other classical groups were studied by Leonard Dickson in the beginning of 20th century. In the 1950s Claude Chevalley
Claude Chevalley
Claude Chevalley was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups...
realized that after an appropriate reformulation, many theorems about semisimple Lie groups admit analogues for algebraic groups over an arbitrary field k, leading to construction of what are now called Chevalley groups. Moreover, as in the case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups (Tits simplicity theorem). Although it was known since 19th century that other finite simple groups exist (for example, Mathieu groups), gradually a belief formed that nearly all finite simple groups can be accounted for by appropriate extensions of Chevalley's construction, together with cyclic and alternating groups. Moreover, the exceptions, the sporadic groups, share many properties with the finite groups of Lie type, and in particular, can be constructed and characterized based on their geometry in the sense of Tits.
The belief has now become a theorem – the classification of finite simple groups
Classification of finite simple groups
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 groups, in much the same way as the prime numbers are the basic...
. Inspection of the list of finite simple groups shows that groups of Lie type over a finite field
Finite field
In 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...
include all the finite simple groups other than the cyclic groups, the alternating groups, 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 ....
, and the 26 sporadic simple groups.
Small groups of Lie type
Many of the smallest groups in the families above have special properties not shared by most members of the family.- Sometimes the smallest groups are solvable rather than simple; for example the groups SL2(2) and SL2(3) are solvable.
- There is a bewildering number of "accidental" isomorphisms between various small groups of Lie type (and alternating groups). For example, the groups SL2(4), PSL2(5), and the alternating group on 5 points are all isomorphic.
- Some of the small groups have a Schur multiplierSchur multiplierIn mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
that is larger than expected. For example, the groups An(q) usually have a Schur multiplier of order (n + 1, q − 1), but the group A2(4) has a Schur multiplier of order 48, instead of the expected value of 3.
For a complete list of these exceptions see the list of finite simple groups.
Many of these special properties are related to certain sporadic simple groups.
Alternating groups sometimes behave as if they were groups of Lie type over the field with one element
Field with one element
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French-English pun, Fun...
.
Some of the small alternating groups also have exceptional properties.
The alternating groups usually have an outer automorphism group
Outer automorphism group
In mathematics, the outer automorphism group of a group Gis the quotient Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out...
of order 2, but the alternating group on 6 points has an outer automorphism group of order 4. Alternating groups usually have a Schur multiplier of order 2, but the ones on 6 or 7 points have a Schur multiplier of order 6.
Notation issues
Unfortunately there is no standard notation for the finite groups of Lie type, and the literature contains dozens of incompatible and confusing systems of notation for them.- The groups of type An−1 are sometimes denoted by PSLn(q) (the projective special linear group) or by Ln(q).
- The groups of type Cn are sometimes denoted by Sp2n(q) (the symplectic group) or (confusingly) by Spn(q).
- The notation for groups of type Dn ("orthogonal" groups) is particularly confusing. Some symbols used are On(q), O−n(q),PSOn(q),
, but there are so many conventions that it is not possible to say exactly what groups these correspond to without it being specified explicitly. The source of the problem is that the simple group is not the orthogonal group O, nor the projective special orthogonal group PSO, but rather a subgroup of PSO, which accordingly does not have a classical notation. A particularly nasty trap is that some authors, such as the ATLASATLAS of Finite GroupsThe ATLAS of Finite Groups, often simply known as the ATLAS, is a group theory book by John Horton Conway, Robert Turner Curtis, Simon Phillips Norton, Richard Alan Parker and Robert Arnott Wilson , published in December 1985 by Oxford University Press and reprinted with corrections in 2003...
, use On(q) for a group that is not the orthogonal group, but the corresponding simple group. The notation
was introduced by Jean DieudonnéJean DieudonnéJean Alexandre Eugène Dieudonné was a French mathematician, notable for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a historian of...
, though his definition is not simple for
and thus the same notation may be used for a slightly different group, which agrees in 
but not in lower dimension.
- For the Steinberg groups, some authors write 2An(q2) (and so on) for the group that other authors denote by 2An(q). The problem is that there are two fields involved, one of order q2, and its fixed field of order q, and people have different ideas on which should be included in the notation. The "2An(q2)" convention is more logical and consistent, but the "2An(q)" convention is far more common and is closer to the convention for algebraic groupAlgebraic groupIn algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s.
- Authors differ on whether groups such as An(q) are the groups of points with values in the simple or the simply connected algebraic group. For example, An(q) may mean either the special linear group SLn+1(q) or the projective special linear group PSLn+1(q). So 2A2(22) may be any one of 4 different groups, depending on the author.