with square the Frobenius exactly when k = 2n + 1 is odd, in which case it is unique. So this gives the finite Ree groups as subgroups of B2(22n+1), F4(22n+1), and G2(32n+1) fixed by an involution. Chevalley groups, Steinberg group, and Ree groups The relation between Chevalley groups, Steinberg group, and Ree groups is roughly as follows. Given a Dynkin diagram X, Chevalley constructed a group scheme over the integers whose values over finite fields are the Chevalley groups. In general one can take the fixed points of an endomorphism of where is the algebraic closure of a finite field, such that some power of is some power of the Frobenius endomorphism φ. The three cases are as follows: For Chevalley groups, for some positive integer n. In this case the group of fixed points is also the group of points of X defined over a finite field. For Steinberg groups, for some positive integers m, n with m dividing n and m > 1. In this case the group of fixed points is also the group of points of a twisted (quasisplit) form of X defined over a finite field. For Ree groups, for some positive integers m, n with m not dividing n. In practice m=2 and n is odd. Ree groups are not given as the points of some connected algebraic group with values in a field. they are the fixed points of an order m=2 automorphism of a group defined over a field of order with n odd, and there is no corresponding field of order pn/2 (although some authors like to pretend there is in their notation for the groups). Ree groups of type 2B2 The Ree groups of type 2B2 were first found by using a different method, and are usually called Suzuki groups. Ree noticed that they could be constructed from the groups of type B2 using a variation of the construction of . Ree realized that a similar construction could be applied to the Dynkin diagrams F4 and G2, leading to two new families of finite simple groups. Ree groups of type 2G2 The Ree groups of type 2G2(32n+1) were introduced by , who showed that they are all simple except for the first one 2G2(3), which is isomorphic to the automorphism group of . gave a simplified construction of the Ree groups, as the automorphisms of a 7-dimensional vector space