In mathematics, a Ree group is a group of Lie type over a finite field constructed by Ree (1960, 1961) from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a different method. They were the last of the infinite families of finite simple groups to be discovered.

Unlike the Steinberg groups, the Ree groups are not given by the points of a connected reductive algebraic group defined over a finite field; in other words, there is no "Ree algebraic group" related to the Ree groups in the same way that (say) unitary groups are related to Steinberg groups. However, there are some exotic pseudo-reductive algebraic groups over non-perfect fields whose construction is related to the construction of Ree groups, as they use the same exotic automorphisms of Dynkin diagrams that change root lengths.

Tits (1960) defined Ree groups over infinite fields of characteristics 2 and 3. Tits (1989) and Hée (1990) introduced Ree groups of infinite-dimensional Kac–Moody algebras.

If X is a Dynkin diagram, Chevalley constructed split algebraic groups corresponding to X, in particular giving groups X(F) with values in a field F. These groups have the following automorphisms:

The Steinberg and Chevalley groups can be constructed as fixed points of an endomorphism of X(F) for F the algebraic closure of a field. For the Chevalley groups, the automorphism is the Frobenius endomorphism of F, while for the Steinberg groups the automorphism is the Frobenius endomorphism times an automorphism of the Dynkin diagram.

Over fields of characteristic 2 the groups B₂(F) and F₄(F) and over fields of characteristic 3 the groups G₂(F) have an endomorphism whose square is the endomorphism α_φ associated to the Frobenius endomorphism φ of the field F. Roughly speaking, this endomorphism α_π comes from the order 2 automorphism of the Dynkin diagram where one ignores the lengths of the roots.

Suppose that the field F has an endomorphism σ whose square is the Frobenius endomorphism: σ² = φ. Then the Ree group is defined to be the group of elements g of X(F) such that α_π(g) = α_σ(g). If the field F is perfect then α_π and α_φ are automorphisms, and the Ree group is the group of fixed points of the involution α_φ/α_π of X(F).

In the case when F is a finite field of order p^k (with p = 2 or 3) there is an endomorphism 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.