Ree group

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.

defined Ree groups over infinite fields of characteristics 2 and 3. and introduced Ree groups of infinite-dimensional Kac–Moody algebras.

Construction

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

• Any endomorphism Template:Mvar of the field Template:Mvar induces an endomorphism Template:Mvar of the group X(F)
• Any automorphism Template:Mvar of the Dynkin diagram induces an automorphism Template:Mvar of the group X(F).

The Steinberg and Chevalley groups can be constructed as fixed points of an endomorphism of X(F) for Template:Mvar the algebraic closure of a field. For the Chevalley groups, the automorphism is the Frobenius endomorphism of Template:Mvar, 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 Template:Mvar associated to the Frobenius endomorphism Template:Mvar of the field Template:Mvar. Roughly speaking, this endomorphism Template:Mvar comes from the order 2 automorphism of the Dynkin diagram where one ignores the lengths of the roots.

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

In the case when Template:Mvar is a finite field of order Template:Mvar (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.

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 Z whose values over finite fields are the Chevalley groups. In general one can take the fixed points of an endomorphism Template:Mvar of X(F) where F is the algebraic closure of a finite field, such that some power of Template:Mvar is some power of the Frobenius endomorphism φ. The three cases are as follows:

• For Chevalley groups, α = φⁿ 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, α^m = φⁿ 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, α^m = φⁿ 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 Template:Mvar with n odd, and there is no corresponding field of order p^n/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 SL₂(8). gave a simplified construction of the Ree groups, as the automorphisms of a 7-dimensional vector space over the field with 3²ⁿ⁺¹ elements preserving a bilinear form, a trilinear form, and a product satisfying a twisted linearity law.

The Ree group has order q³(q³ + 1)(q − 1) where q = 3²ⁿ⁺¹

The Schur multiplier is trivial for n ≥ 1 and for ²G₂(3)′.

The outer automorphism group is cyclic of order 2n + 1.

The Ree group is also occasionally denoted by Ree(q), R(q), or E₂^*(q)

The Ree group ²G₂(q) has a doubly transitive permutation representation on q³ + 1 points, and more precisely acts as automorphisms of an S(2, q+1, q³+1) Steiner system. It also acts on a 7-dimensional vector space over the field with q elements as it is a subgroup of G₂(q).

The 2-sylow subgroups of the Ree groups are elementary abelian of order 8. Walter's theorem shows that the only other non-abelian finite simple groups with abelian Sylow 2-subgroups are the projective special linear groups in dimension 2 and the Janko group J1. These groups also played a role in the discovery of the first modern sporadic group. They have involution centralizers of the form Z/2Z × PSL₂(q), and by investigating groups with an involution centralizer of the similar form Z/2Z × PSL₂(5) Janko found the sporadic group J₁. determined their maximal subgroups.

The Ree groups of type ²G₂ are exceptionally hard to characterize. Thompson (1967, 1972, 1977) studied this problem, and was able to show that the structure of such a group is determined by a certain automorphism Template:Mvar of a finite field of characteristic 3, and that if the square of this automorphism is the Frobenius automorphism then the group is the Ree group. He also gave some complicated conditions satisfied by the automorphism Template:Mvar. Finally Bombieri (1980) used elimination theory to show that Thompson's conditions implied that σ² = 3 in all but 178 small cases, that were eliminated using a computer by Odlyzko and Hunt. Bombieri found out about this problem after reading an article about the classification by , who suggested that someone from outside group theory might be able to help solving it. gave a unified account of the solution of this problem by Thompson and Bombieri.

Ree groups of type ²F₄

The Ree groups of type ²F₄(2²ⁿ⁺¹) were introduced by . They are simple except for the first one ²F₄(2), which showed has a simple subgroup of index 2, now known as the Tits group. gave a simplified construction of the Ree groups as the symmetries of a 26-dimensional space over the field of order 2²ⁿ⁺¹ preserving a quadratic form, a cubic form, and a partial multiplication.

The Ree group ²F₄(2²ⁿ⁺¹) has order q¹²(q⁶ + 1) (q⁴ − 1) (q³ + 1) (q − 1) where q = 2²ⁿ⁺¹. The Schur multiplier is trivial. The outer automorphism group is cyclic of order 2n + 1.

These Ree groups have the unusual property that the Coxeter group of their BN pair is not crystallographic: it is the dihedral group of order 16. showed that all Moufang octagons come from Ree groups of type ²F₄.

See also

• List of finite simple groups

External links

• ATLAS: Ree group R(27)