Group of Lie type

In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but the important collection of finite simple 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, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but the important collection of finite simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups. The name 'groups of Lie type' is due to the close relationship with the (infinite) Lie groups, since a compact Lie group may be viewed as the rational points of a reductive linear algebraic group over the field of real numbers. Dieudonné (1971) and Carter (1989) are standard references for groups of Lie type. An initial approach to this question was the definition and detailed study of the so-called classical groups over finite and other fields by Jordan (1870). 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 quotients, the latter yielding projective linear groups. 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 can be thought of as Lie groups over finite fields. The theory was clarified by the theory of algebraic groups, and the work of Chevalley (1955) on Lie algebras, by means of which the Chevalley group concept was isolated. Chevalley constructed a Chevalley basis (a sort of integral form but over finite fields) 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 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 (sometimes called Dickson groups) had already been constructed by Dickson (1905), and the ones of type E6 by Dickson (1901). Chevalley's construction did not give all of the known classical groups: it omitted the unitary groups and the non-split orthogonal groups. Steinberg (1959) 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 Tits (1958). 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 numbers 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.