Ree group

In mathematics, aRee groupis agroup of Lie typeover afinite fieldconstructed byRee(1960,1961) from an exceptionalautomorphismof aDynkin diagramthat reverses the direction of the multiple bonds, generalizing theSuzuki groupsfound by Suzuki using a different method. They were the last of the infinite families offinite simple groupsto be discovered.

Unlike theSteinberg groups,the Ree groups are not given by the points of a connectedreductive algebraic groupdefined 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 exoticpseudo-reductive algebraic groupsover 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)andHée (1990)introduced Ree groups of infinite-dimensionalKac–Moody algebras.

Construction

If $X$ is aDynkin 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:

Any endomorphism $σ$ of the field $F$ induces an endomorphism $α σ$ of the group $X (F)$
Any automorphism $π$ of the Dynkin diagram induces an automorphism $α π$ of the group $X (F)$ .

The Steinberg and Chevalley groups can be constructed as fixed points of an endomorphism ofX(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 $B 2 (F)$ and $F 4 (F)$ and over fields of characteristic 3 the groups $G 2 (F)$ 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: $σ 2 = φ$ .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 $p k$ (withp= 2 or 3) there is an endomorphism with square the Frobenius exactly whenk= 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 diagramX,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 $α$ of $X (F)$ where $F$ is the algebraic closure of a finite field, such that some power of $α$ is some power of the Frobenius endomorphism φ. The three cases are as follows:

For Chevalley groups, $α = φ n$ for some positive integern.In this case the group of fixed points is also the group of points ofXdefined over a finite field.
For Steinberg groups, $α m = φ n$ for some positive integersm,nwithmdividingnandm> 1. In this case the group of fixed points is also the group of points of a twisted (quasisplit) form ofXdefined over a finite field.
For Ree groups, $α m = φ n$ for some positive integersm,nwithmnot dividingn.In practicem=2 andnis 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 orderm=2 automorphism of a group defined over a field of order $p n$ withnodd, and there is no corresponding field of orderp^n/2(although some authors like to pretend there is in their notation for the groups).

Ree groups of type²B₂

The Ree groups of type²B₂were first found bySuzuki (1960)using a different method, and are usually calledSuzuki groups.Ree noticed that they could be constructed from the groups of type B₂using a variation of the construction ofSteinberg (1959).Ree realized that a similar construction could be applied to the Dynkin diagrams F₄and G₂,leading to two new families of finite simple groups.

Ree groups of type²G₂

The Ree groups of type²G₂(3²ⁿ⁺¹) were introduced byRee (1960),who showed that they are all simple except for the first one²G₂(3), which is isomorphic to the automorphism group of $SL 2 (8)$ .Wilson (2010)gave a simplified construction of the Ree groups, as the automorphisms of a 7-dimensional vector space over the field with 3²ⁿ⁺¹elements preserving a bilinear form, a trilinear form, and a product satisfying a twisted linearity law.

The Ree group has order $q 3 (q 3 + 1)(q - 1)$ whereq= 3²ⁿ⁺¹

The Schur multiplier is trivial forn≥ 1 and for²G₂(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 group²G₂(q) has adoubly transitive permutation representationon $q 3 + 1$ 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 withqelements as it is a subgroup of G₂(q).

The 2-sylow subgroups of the Ree groups are elementary abelian of order 8.Walter's theoremshows that the only other non-abelian finite simple groups with abelian Sylow 2-subgroups are the projective special linear groups in dimension 2 and theJanko group J1.These groups also played a role in the discovery of the first modern sporadic group. They have involution centralizers of the form $Z /2 Z \times PSL 2 (q)$ ,and by investigating groups with an involution centralizer of the similar form $Z /2 Z \times PSL 2 (5)$ Janko found the sporadic groupJ₁.Kleidman (1988)determined their maximal subgroups.

The Ree groups of type²G₂are 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 $σ$ 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 $σ$ .Finally Bombieri (1980) usedelimination theoryto show that Thompson's conditions implied that $σ 2 = 3$ in all but 178 small cases, that were eliminated using a computer byOdlyzkoand Hunt. Bombieri found out about this problem after reading an article about the classification byGorenstein (1979),who suggested that someone from outside group theory might be able to help solving it.Enguehard (1986)gave a unified account of the solution of this problem by Thompson and Bombieri.

Ree groups of type²F₄

The Ree groups of type $2 F 4 (2 2 n +1)$ were introduced byRee (1961).They are simple except for the first one $2 F 4 (2)$ ,whichTits (1964)showed has a simple subgroup of index 2, now known as theTits group. Wilson (2010b)gave a simplified construction of the Ree groups as the symmetries of a 26-dimensional space over the field of order 2²ⁿ⁺¹preserving a quadratic form, a cubic form, and a partial multiplication.

The Ree group $2 F 4 (2 2 n +1)$ has order q¹²(q⁶+ 1) (q⁴− 1) (q³+ 1) (q− 1) where q= 2²ⁿ⁺¹. TheSchur multiplieris trivial. Theouter automorphism groupis cyclic of order 2n+ 1.

These Ree groups have the unusual property that theCoxeter groupof theirBN pairis not crystallographic: it is the dihedral group of order 16.Tits (1983)showed that allMoufang octagonscome from Ree groups of type $2 F 4$ .

References

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Bombieri, Enrico(1980), "Thompson's problem (σ²=3) ",Inventiones Mathematicae,58(1), appendices by Andrew Odlyzko and D. Hunt: 77–100,doi:10.1007/BF01402275,ISSN 0020-9910,MR 0570875,S2CID 122867511
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Hée, Jean-Yves (1990), "Construction de groupes tordus en théorie de Kac-Moody",Comptes Rendus de l'Académie des Sciences, Série I,310(3): 77–80,ISSN 0764-4442,MR 1044619
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External links

ATLAS: Ree group R(27)

Ree group

Contents

Construction

Chevalley groups, Steinberg group, and Ree groups

Ree groups of type²B₂

Ree groups of type²G₂

Ree groups of type²F₄

See also

References

External links

Ree group

Construction

Chevalley groups, Steinberg group, and Ree groups

Ree groups of type2B2

Ree groups of type2G2

Ree groups of type2F4

See also

References

External links

Ree groups of type²B₂

Ree groups of type²G₂

Ree groups of type²F₄