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).
Emil Artin investigated the orders of such groups, with a view to classifying cases of coincidence.
There are several minor variations of these, given by taking derived subgroups or central quotients, the latter yielding projective linear groups.
They correspond to the series An, Bn, Cn, Dn,2An, 2Dn of Chevalley and Steinberg 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).
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.
Analogously to the unitary case, Steinberg constructed families of groups by taking fixed points of a product of a diagram and a field automorphism.
These gave: The groups of type 3D4 have no analogue over the reals, as the complex numbers have no automorphism of order 3.
[clarification needed] The symmetries of the D4 diagram also give rise to triality.
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.
Ree was able to find two new similar families and of simple groups by using the fact that F4 and G2 have extra automorphisms in characteristic 2 and 3.
However some of the smallest groups in the families above are either not perfect or have a Schur multiplier larger than "expected".