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In mathematics, a group of Lie type is a finite group related to the points of a simple algebraic group with values in a finite field. The classification of finite simple groups shows that they include all the finite simple groups other than the cyclic groups, the alternating groups, the Tits group, and the 26 sporadic simple groups. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki-Ree groups. Mathematics is the study of quantity, structure, space and change. ...
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. ...
In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...
The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ...
In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ...
In mathematics an alternating group is the group of even permutations of a finite set. ...
The Tits group 2F4(2) is a finite simple group of order 17971200 named for the Belgian mathematician Jacques Tits. ...
The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ...
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. Much work was done on this, from the time of L. E. Dickson to the book of Jean Dieudonné. For example Emil Artin investigated the orders of such groups, with a view to classifying cases of coincidence. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
Emil Artin (March 3, 1898-December 20, 1962) was a mathematician born in Vienna, Austria who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ...
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 quotienting out by the center. 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. In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ...
In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ...
In mathematics, the unitary group of degree n over the field F (which is either the field R of real numbers or the field C of complex numbers) is the group of n by n unitary matrices with entries from F, with the group operation that of matrix multiplication. ...
In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if...
Chevalley groups The theory was clarified by the theory of algebraic groups, and the work of Claude Chevalley in the mid-1950s on the Lie algebras by means of which the Chevalley group concept was isolated. Chevalley constructed integral forms of all the complex simple Lie algebras (or rather of their universal enveloping algebras), 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. (Some of these has already been constructed by Dickson.) 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. ...
Claude Chevalley (11 February 1909 - 28 June 1984) was a French mathematician with an austere style based on abstract algebra. ...
In mathematics, a group of Lie type is a finite group related to the points of a simple algebraic group with values in a finite field. ...
In mathematics, a simple Lie group is a Lie group which is also a simple group. ...
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). ...
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. Steinberg found a modification of Chevalley's construction that gave these groups and a few new families. His construction is similar to the usual construction of the unitary group from the general linear group. The general linear group over the complex numbers has a "diagram automorphism" given by taking the transpose inverse, and a "field automorphism" given by taking complex conjugation. 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. 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 coming from the order 2 automorphism of An, some more orthogonal groups 2Dn from the order 2 automorphism of Dn, and 2 new series 2E6, 3D4 from the automorphisms of order 2 and 3 of E6 and D4. (The groups of type 3D4 have no analogue over the reals, as the complex numbers have no automorphism of order 3.) See also Simple Lie group. ...
Suzuki-Ree groups Around 1960, Suzuki caused a sensation by finding a new infinite series of groups that at first sight seemed unrelated to the known algebraic groups. Ree 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 In mathematics, the Frobenius automorphism is an automorphism induced by a prime power mapping defined for various extensions of fields. ...
- 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 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 2, called the Tits group (named after the mathematician Jacques Tits). The smallest group 3G2(3) of type 3G2 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 Jacques Tits (born August 12, 1930) is a Belgian mathematician. ...
The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple 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. In mathematics, the Janko groups J1, J2, J3 and J4 are four of the twenty-six sporadic groups; their respective orders are: J1 The smallest Janko group, J1 of order 175560, has a presentation in terms of two generators a and b and c = abab-1 as It can also...
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 multiplier 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. The existence of these 'small' phenomena is not entirely a matter of 'trivia'; they are reflected elsewhere, for example in homotopy theory. In group theory, the Schur multiplier is the second homology group of a group G with coefficients in the integers, . If the group is presented in terms of a free group F on a set of generators, and a normal subgroup R generated by a set of relations on the...
In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type (including the Tits group, which strictly speaking is not of Lie type), or one of 26 sporadic groups. ...
An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...
Alternating groups sometimes behave as if they were groups of Lie type over the (non-existent) field with 1 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.
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, some of which could hardly be worse had they been specifically designed to confuse newcomers. - 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 by Spn(q) (in the case of particularly evil minded authors).
- The notation for orthogonal groups is particularly confusing. Some symbols used are On(q), O−n(q),PSOn(q), Ωn(q), but there are so many conventions that it is not possible to say exactly what groups these correspond to. A particularly nasty trap is that some authors use On(q) for a group that is not the orthogonal group, but the corresponding simple group.
- 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.
- 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.
Further reading A standard reference is - Simple Groups of Lie Type by Roger W. Carter, ISBN 0471506834
The classical groups are described in - La géométrie des groupes classiques by Jean Dieudonné
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