Group of Lie type

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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 finite simple groups. Special cases include the classical groups, the Chevalley groups, the Steinberg groups, and the Suzuki-Ree groups.

Contents 1 Classical groups 2 Chevalley groups 3 Steinberg groups 4 Suzuki-Ree groups 5 Relations with finite simple groups 6 Small groups of Lie type 7 Notation issues 8 Further reading

[edit] 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.

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 A_n, B_n, C_n, D_n, ²A_n, ²D_n of Chevalley and Steinberg groups.

[edit] 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 a Chevalley basis (a sort of integral form) for 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 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. (Some of these had already been constructed by Dickson.)

[edit] 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 ²A_n coming from the order 2 automorphism of A_n, some more orthogonal groups ²D_n from the order 2 automorphism of D_n, 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.)

[edit] Suzuki-Ree groups

Around 1960, Michio 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

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

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.

[edit] Relations with finite simple groups

Finite groups of Lie type were among the first groups to be considered in mathematics, after cyclic, symmetric and alternating groups. Their exploration started with 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 PSL_n(q) of finite simple groups. Other classical groups were studied by Leonard Dickson in the beginning of 20th century. In the 1950's 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 is know 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. 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.

[edit] 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 A_n(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.

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.

[edit] 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 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_2n(q) (the symplectic group) or by Sp_n(q) (in the case of particularly evil minded authors).

• The notation for 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. A particularly nasty trap is that some authors use O_n(q) for a group that is not the orthogonal group, but the corresponding simple group.

• For the Steinberg groups, some authors write ²A_n(q²) (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 q², and its fixed field of order q, and people have different ideas on which should be included in the notation. The "²A_n(q²)" convention is more logical and consistent, but the "²A_n(q)" convention is far more common.

• 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(2²) may be any one of 4 different groups, depending on the author.

[edit] Further reading

A standard reference is

• Simple Groups of Lie Type by Roger W. Carter, ISBN 0-471-50683-4

The classical groups are described in

• La géométrie des groupes classiques by Jean Dieudonné