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.

= Classical groups =

An initial approach to this question was the definition and detailed study of the so-called classical groups over finite and other field (mathematics). 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 group, orthogonal group, symplectic group, or unitary group group. There are several minor variations of these, given by taking derived subgroups or quotienting out by the center of a group. 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 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.)

= 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.)

= 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 B 2 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 22 n +1, and the corresponding groups are the Suzuki groups

:2 B 2(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 of order 20.) Ree was able to find two new similar families

:2 F 4(22 n +1)

and

:2 G 2(32 n +1)

of simple groups by using the fact that F4 and G 2 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 2 F 4(2) of type 2 F 4 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 3 G 2(3) of type 3 G 2 is not simple, but it has a simple normal subgroup of index 3, isomorphic to SL 2(8). In the classification of finite simple groups, the Ree groups

:2 G 2(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 /2 Z  ×  PSL 2( q ) for q = 3 n , and by investigating groups with an involution centralizer of the similar form Z /2 Z  ×  PSL 2(5) Janko found the sporadic group Janko group.

= 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 SL 2(2) and SL 2(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 SL 2(4), PSL 2(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 A 2(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.

= 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.

= Further reading =

A standard reference is

The classical groups are described in