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In mathematics, specifically in
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the phrase ''group of Lie type'' usually refers to
finite groups Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
that are closely related to the group of
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
s of a reductive
linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...
with values in a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
. The phrase ''group of Lie type'' does not have a widely accepted precise definition,mathoverflow - Definition of “finite group of Lie type”?
/ref> but the important collection of finite
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
groups of Lie type does have a precise definition, and they make up most of the groups in the
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
. The name "groups of Lie type" is due to the close relationship with the (infinite)
Lie groups In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additi ...
, since a
compact Lie group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
may be viewed as the rational points of a reductive linear algebraic group over the field of real numbers. 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
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
by . These groups were studied by
L. E. Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also reme ...
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 In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
. There are several minor variations of these, given by taking
derived subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest norma ...
s or central
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
s, the latter yielding
projective linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
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 A''n'', B''n'', C''n'', D''n'',2A''n'', 2D''n'' of Chevalley and Steinberg groups.


Chevalley groups

Chevalley groups can be thought of as Lie groups over finite fields. The theory was clarified by the theory of
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
s, and the work of on Lie algebras, by means of which the ''Chevalley group'' concept was isolated. Chevalley constructed a
Chevalley basis In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite f ...
(a sort of integral form but over finite fields) for all the complex
simple Lie algebra In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of s ...
s (or rather of their
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the represent ...
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 A''n'', B''n'', C''n'', D''n'' 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 (sometimes called ''Dickson groups'') 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 group In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''- dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called the ...
s. 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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s has a ''diagram automorphism'' given by reversing the
Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
A''n'' (which corresponds to taking the transpose inverse), and a ''
field automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
'' 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'' 2A''n'', from the order 2 automorphism of A''n''; * further ''orthogonal groups'' 2D''n'', from the order 2 automorphism of D''n''; * 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 D4 diagram also give rise to
triality In mathematics, triality is a relationship among 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( ...
.


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 In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...
. 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 22''n''+1, and the corresponding groups are the Suzuki groups :2B2(22''n''+1) = Suz(22''n''+1). (Strictly speaking, the group Suz(2) is not counted as a Suzuki group as it is not simple: it is the
Frobenius group In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius. Structure Suppos ...
of order 20.) Ree was able to find two new similar families :2F4(22''n''+1) and :2G2(32''n''+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 2, called the ''
Tits group In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order :   211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a 27th sporadic group ...
'' (named after the mathematician
Jacques Tits Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life an ...
). The smallest group 2G2(3) of type 2G2 is not simple, but it has a simple normal subgroup of index 3, isomorphic to A1(8). In the
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
, the Ree groups :2G2(32''n''+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 × PSL(2, ''q'') for ''q'' = 3''n'', and by investigating groups with an involution centralizer of the similar form Z/2Z × PSL(2, 5) Janko found the sporadic group  ''J''1. The Suzuki groups are the only finite non-abelian simple groups with order not divisible by 3. They have order 22(2''n''+1)(22(2''n''+1) + 1)(2(2''n''+1) − 1).


Relations with finite simple groups

Finite groups of Lie type were among the first groups to be considered in mathematics, after
cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...
,
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and alternating groups, with the projective special linear groups over prime finite fields, PSL(2, ''p'') being constructed by
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
in the 1830s. The systematic exploration of finite groups of Lie type started with
Camille Jordan Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at ...
's theorem that the projective special linear group PSL(2, ''q'') is simple for ''q'' ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL(''n'', ''q'') of finite simple groups. Other classical groups were studied by
Leonard Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also reme ...
in the beginning of 20th century. In the 1950s Claude Chevalley realized that after an appropriate reformulation, many theorems about
semisimple Lie group In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
s 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 In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 object ...
), 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 In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. Th ...
, 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 In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
. Inspection of the list of finite simple groups shows that groups of Lie type over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
include all the finite simple groups other than the cyclic groups, the alternating groups, the
Tits group In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order :   211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a 27th sporadic group ...
, and the 26 sporadic simple groups.


Small groups of Lie type

In general the finite group associated to an endomorphism of a simply connected simple algebraic group is the universal central extension of a simple group, so is perfect and has trivial Schur multiplier. However some of the smallest groups in the families above are either not perfect or have a Schur multiplier larger than "expected". Cases where the group is not perfect include *A1(2) = SL(2, 2) Solvable of order 6 (the symmetric group on 3 points) *A1(3) = PSL(2, 3) Solvable of order 12 (the alternating group on 4 points) *2A2(4) Solvable *B2(2) Not perfect, but is isomorphic to the symmetric group on 6 points so its derived subgroup has index 2 and is simple of order 360. *2B2(2) = Suz(2) Solvable of order 20 (a Frobenius group) *2F4(2) Not perfect, but the derived group has index 2 and is the simple
Tits group In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of order :   211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a 27th sporadic group ...
. *G2(2) Not perfect, but the derived group has index 2 and is simple of order 6048. *2G2(3) Not perfect, but the derived group has index 3 and is the simple group of order 504. Some cases where the group is perfect but has a Schur multiplier that is larger than expected include: *A1(4) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1. *A1(9) The Schur multiplier has an extra Z/3Z, so the Schur multiplier of the simple group has order 6 instead of 2. *A2(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1. *A2(4) The Schur multiplier has an extra Z/4Z × Z/4Z, so the Schur multiplier of the simple group has order 48 instead of 3. *A3(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1. *B3(2) = C3(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1. *B3(3) The Schur multiplier has an extra Z/3Z, so the Schur multiplier of the simple group has order 6 instead of 2. *D4(2) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order 4 instead of 1. *F4(2) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1. *G2(3) The Schur multiplier has an extra Z/3Z, so the Schur multiplier of the simple group has order 3 instead of 1. *G2(4) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1. *2A3(4) The Schur multiplier has an extra Z/2Z, so the Schur multiplier of the simple group has order 2 instead of 1. *2A3(9) The Schur multiplier has an extra Z/3Z × Z/3Z, so the Schur multiplier of the simple group has order 36 instead of 4. *2A5(4) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order 12 instead of 3. *2E6(4) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order 12 instead of 3. *2B2(8) The Schur multiplier has an extra Z/2Z × Z/2Z, so the Schur multiplier of the simple group has order 4 instead of 1. There is a bewildering number of "accidental" isomorphisms between various small groups of Lie type (and alternating groups). For example, the groups SL(2, 4), PSL(2, 5), and the alternating group on 5 points are all isomorphic. For a complete list of these exceptions see the
list of finite simple groups A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby unio ...
. 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 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. The name ...
. Some of the small alternating groups also have exceptional properties. The alternating groups usually have an
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
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

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 simple group PSL(''n'', ''q'') is not usually the same as the group PSL(''n'', F''q'') of F''q''-valued points of the algebraic group PSL(''n''). The problem is that a surjective map of algebraic groups such as SL(''n'') → PSL(''n'') does not necessarily induce a surjective map of the corresponding groups with values in some (non algebraically closed) field. There are similar problems with the points of other algebraic groups with values in finite fields. * The groups of type A''n''−1 are sometimes denoted by PSL(''n'', ''q'') (the projective special linear group) or by ''L''(''n'', ''q''). * The groups of type C''n'' are sometimes denoted by Sp(2''n'', ''q'') (the symplectic group) or (confusingly) by Sp(''n'', ''q''). * The notation for groups of type D''n'' ("orthogonal" groups) is particularly confusing. Some symbols used are O(''n'', ''q''), ''O''(''n'', ''q''), PSO(''n'', ''q''), Ω''n''(''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,
ATLAS An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geograp ...

p. xi
/ref> which accordingly does not have a classical notation. A particularly nasty trap is that some authors, such as the
ATLAS An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geograp ...
, use O(''n'', ''q'') for a group that is ''not'' the orthogonal group, but the corresponding simple group. The notation Ω, PΩ was introduced by Jean Dieudonné, though his definition is not simple for ''n'' ≤ 4 and thus the same notation may be used for a slightly different group, which agrees in ''n'' ≥ 5 but not in lower dimension. * For the Steinberg groups, some authors write 2A''n''(''q''2) (and so on) for the group that other authors denote by 2A''n''(''q''). The problem is that there are two fields involved, one of order ''q''2, and its fixed field of order ''q'', and people have different ideas on which should be included in the notation. The "2A''n''(''q''2)" convention is more logical and consistent, but the "2A''n''(''q'')" convention is far more common and is closer to the convention for
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
s. * Authors differ on whether groups such as A''n''(''q'') are the groups of points with values in the simple or the simply connected algebraic group. For example, A''n''(''q'') may mean either the special linear group SL(''n''+1, ''q'') or the projective special linear group PSL(''n''+1, ''q''). So 2A2(4) may be any one of 4 different groups, depending on the author.


See also

* Deligne–Lusztig theory (
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of finite groups of Lie type) * Modular Lie algebra


Notes


References

* * * * * Leonard E. Dickson reported groups of type G2 * * * * * * * * {{DEFAULTSORT:Group Of Lie Type Group theory Lie algebras Algebraic groups