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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 ''An'', ''Bn'', ''Cn'', ''Dn'', ''2An'', ''2Dn'' of Chevalley and Steinberg groups. CHEVALLEY GROUPS The theory was clarified by the theory of Algebraic Group s, 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 Algebra s (or rather of their Universal Enveloping Algebra 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 ''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 had 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 Number s 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 Diagram s, 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, 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 ''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 2''G''2(3) of type 2''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 ''J''1 . 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 Camille Jordan '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 in the beginning of 20th century. In the 1950s Claude Chevalley realized that after an appropriate reformulation, many theorems about Semisimple Lie Group s admit analogues for algebraic groups over an arbitrary field ''k'', leading to construction of what is 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 ), 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 Group s. 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.
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.
SEE ALSO
REFERENCES A standard reference is
The classical groups are described in
Leonard E. Dickson reported groups of type G2 in
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