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
editIfXis aDynkin diagram,Chevalley constructed split algebraic groups corresponding toX,in particular giving groupsX(F)with values in a fieldF.These groups have the following automorphisms:
- Any endomorphismσof the fieldFinduces an endomorphismασof the groupX(F)
- Any automorphismπof the Dynkin diagram induces an automorphismαπof the groupX(F).
The Steinberg and Chevalley groups can be constructed as fixed points of an endomorphism ofX(F) forFthe algebraic closure of a field. For the Chevalley groups, the automorphism is the Frobenius endomorphism ofF,while for the Steinberg groups the automorphism is the Frobenius endomorphism times an automorphism of the Dynkin diagram.
Over fields of characteristic 2 the groupsB2(F)andF4(F)and over fields of characteristic 3 the groupsG2(F)have an endomorphism whose square is the endomorphismαφassociated to the Frobenius endomorphismφof the fieldF.Roughly speaking, this endomorphismαπcomes from the order 2 automorphism of the Dynkin diagram where one ignores the lengths of the roots.
Suppose that the fieldFhas an endomorphismσwhose square is the Frobenius endomorphism:σ2=φ.Then the Ree group is defined to be the group of elementsgofX(F)such thatαπ(g) =ασ(g).If the fieldFis perfect thenαπandαφare automorphisms, and the Ree group is the group of fixed points of the involutionαφ/απofX(F).
In the case whenFis a finite field of orderpk(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
editThe relation between Chevalley groups, Steinberg group, and Ree groups is roughly as follows. Given a Dynkin diagramX,Chevalley constructed a group scheme over the integersZwhose values over finite fields are the Chevalley groups. In general one can take the fixed points of an endomorphismαofX(F)whereFis the algebraic closure of a finite field, such that some power ofαis some power of the Frobenius endomorphism φ. The three cases are as follows:
- For Chevalley groups,α=φnfor 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,αm=φnfor 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,αm=φnfor 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 orderpnwithnodd, and there is no corresponding field of orderpn/2(although some authors like to pretend there is in their notation for the groups).
Ree groups of type2B2
editThe 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
editThe 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 ofSL2(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 orderq3(q3+ 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 representationonq3+ 1points, 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 formZ/2Z× PSL2(q),and by investigating groups with an involution centralizer of the similar formZ/2Z× PSL2(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 automorphismσof 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 automorphismσ.Finally Bombieri (1980) usedelimination theoryto show that Thompson's conditions implied thatσ2= 3in 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
editThe Ree groups of type2F4(22n+1)were introduced byRee (1961).They are simple except for the first one2F4(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 group2F4(22n+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 type2F4.
See also
editReferences
edit- Carter, Roger W.(1989) [1972],Simple groups of Lie type,Wiley Classics Library, New York:John Wiley & Sons,ISBN978-0-471-50683-6,MR0407163
- 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,ISSN0020-9910,MR0570875,S2CID122867511
- Enguehard, Michel (1986), "Caractérisation des groupes de Ree",Astérisque(142): 49–139,ISSN0303-1179,MR0873958
- Gorenstein, D.(1979), "The classification of finite simple groups. I. Simple groups and local analysis",Bulletin of the American Mathematical Society,New Series,1(1): 43–199,doi:10.1090/S0273-0979-1979-14551-8,ISSN0002-9904,MR0513750
- 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,ISSN0764-4442,MR1044619
- Kleidman, Peter B. (1988), "The maximal subgroups of the Chevalley groups G2(q) with q odd, the Ree groups2G2(q), and their automorphism groups ",Journal of Algebra,117(1): 30–71,doi:10.1016/0021-8693(88)90239-6,ISSN0021-8693,MR0955589
- Ree, Rimhak (1960),"A family of simple groups associated with the simple Lie algebra of type (G2) ",Bulletin of the American Mathematical Society,66(6): 508–510,doi:10.1090/S0002-9904-1960-10523-X,ISSN0002-9904,MR0125155
- Ree, Rimhak (1961),"A family of simple groups associated with the simple Lie algebra of type (F4) ",Bulletin of the American Mathematical Society,67:115–116,doi:10.1090/S0002-9904-1961-10527-2,ISSN0002-9904,MR0125155
- Steinberg, Robert (1959),"Variations on a theme of Chevalley",Pacific Journal of Mathematics,9(3): 875–891,doi:10.2140/pjm.1959.9.875,ISSN0030-8730,MR0109191
- Steinberg, Robert (1968),Lectures on Chevalley groups,Yale University, New Haven, Conn.,MR0466335,archived fromthe originalon 2012-09-10
- Steinberg, Robert (1968),Endomorphisms of linear algebraic groups,Memoirs of the American Mathematical Society, No. 80, Providence, R.I.:American Mathematical Society,ISBN9780821812808,MR0230728
- Suzuki, Michio(1960), "A new type of simple groups of finite order",Proceedings of the National Academy of Sciences of the United States of America,46(6): 868–870,doi:10.1073/pnas.46.6.868,ISSN0027-8424,JSTOR70960,MR0120283,PMC222949,PMID16590684
- Thompson, John G.(1967), "Toward a characterization of E2*(q) ",Journal of Algebra,7(3): 406–414,doi:10.1016/0021-8693(67)90080-4,ISSN0021-8693,MR0223448
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- Thompson, John G.(1977), "Toward a characterization of E2*(q). III ",Journal of Algebra,49(1): 162–166,doi:10.1016/0021-8693(77)90276-9,ISSN0021-8693,MR0453858
- Tits, Jacques (1960),"Les groupes simples de Suzuki et de Ree",Séminaire Bourbaki, Vol. 6,Paris:Société Mathématique de France,pp. 65–82,MR1611778
- Tits, Jacques (1964), "Algebraic and abstract simple groups",Annals of Mathematics,Second Series,80(2): 313–329,doi:10.2307/1970394,ISSN0003-486X,JSTOR1970394,MR0164968
- Tits, Jacques (1983), "Moufang octagons and the Ree groups of type2F4",American Journal of Mathematics,105(2): 539–594,doi:10.2307/2374268,ISSN0002-9327,JSTOR2374268,MR0701569
- Tits, Jacques (1989),"Groupes associés aux algèbres de Kac-Moody",Astérisque,Séminaire Bourbaki (177): 7–31,ISSN0303-1179,MR1040566
- Wilson, Robert A. (2010), "Another new approach to the small Ree groups",Archiv der Mathematik,94(6): 501–510,CiteSeerX10.1.1.156.9909,doi:10.1007/s00013-010-0130-4,ISSN0003-9268,MR2653666,S2CID122724281
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