20 - Suzuki and Ree Groups

Summary

In this chapter we look at the simple groups found by Ree and Suzuki, which exist only over fields of characteristic 2 or 3. The Suzuki groups are subgroups of an algebraic group of type B₂ (= C₂) when p = 2, while the two families of Ree groups occur in algebraic groups of types G₂ when p = 3 and F4 when p = 2. The idea is to combine Chevalley's special isogeny (5.3) with a standard Frobenius map, then take fixed points.

In contrast to the theory for Chevalley groups and their twisted analogues of types A,D,E₆, the representation theory of these groups in the defining characteristic p = 2 or p = 3 involves just one fixed prime and therefore does not raise the genericity questions considered earlier. On the other hand, there is still much to be made explicit as the power of p grows.

Here we review the literature dealing with Suzuki and Ree groups, giving some examples and indicating some unsolved problems. After recalling how the groups are constructed (20.1), we discuss their simple modules (20.2). Beyond this, the treatment of projective modules, Cartan invariants, extensions, and cohomology is somewhat fragmentary. The Suzuki groups have been best studied. Similarly, while all the ordinary characters have been determined (20.7), their reduction modulo p has not been worked out in detail for Ree groups.