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.

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.

Construction

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:

• 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 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.

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 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 = 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 pⁿ 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

Main article: Suzuki groups

The Ree groups of type ²B₂ were first found by Suzuki (1960) 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 Steinberg (1959). 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 Ree (1960), who showed that they are all simple except for the first one ²G₂(3), which is isomorphic to the automorphism group of SL₂(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⁺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. Kleidman (1988) 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 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) 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 Gorenstein (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 type ²F₄

The Ree groups of type ²F₄(2²ⁿ⁺¹) were introduced by Ree (1961). They are simple except for the first one ²F₄(2), which Tits (1964) showed has a simple subgroup of index 2, now known as the Tits 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⁺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. Tits (1983) showed that all Moufang octagons come from Ree groups of type ²F₄.

See also

• List of finite simple groups

References

• Carter, Roger W. (1989) [1972], Simple groups of Lie type, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-50683-6, MR 0407163
• 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
• Enguehard, Michel (1986), "Caractérisation des groupes de Ree", Astérisque (142): 49–139, ISSN 0303-1179, MR 0873958
• 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, ISSN 0002-9904, MR 0513750
• 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
• Kleidman, Peter B. (1988), "The maximal subgroups of the Chevalley groups G₂(q) with q odd, the Ree groups ²G₂(q), and their automorphism groups", Journal of Algebra, 117 (1): 30–71, doi:10.1016/0021-8693(88)90239-6, ISSN 0021-8693, MR 0955589
• 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, ISSN 0002-9904, MR 0125155
• 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, ISSN 0002-9904, MR 0125155
• Steinberg, Robert (1959), "Variations on a theme of Chevalley", Pacific Journal of Mathematics, 9 (3): 875–891, doi:10.2140/pjm.1959.9.875, ISSN 0030-8730, MR 0109191
• Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335, archived from the original on 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, ISBN 9780821812808, MR 0230728
• 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, ISSN 0027-8424, JSTOR 70960, MR 0120283, PMC 222949, PMID 16590684
• 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, ISSN 0021-8693, MR 0223448
• Thompson, John G. (1972), "Toward a characterization of E₂*(q) . II", Journal of Algebra, 20 (3): 610–621, doi:10.1016/0021-8693(72)90074-9, ISSN 0021-8693, MR 0313377
• 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, ISSN 0021-8693, MR 0453858
• Tits, Jacques (1960), "Les groupes simples de Suzuki et de Ree", Séminaire Bourbaki, Vol. 6, Paris: Société Mathématique de France, pp. 65–82, MR 1611778
• Tits, Jacques (1964), "Algebraic and abstract simple groups", Annals of Mathematics, Second Series, 80 (2): 313–329, doi:10.2307/1970394, ISSN 0003-486X, JSTOR 1970394, MR 0164968
• Tits, Jacques (1983), "Moufang octagons and the Ree groups of type ²F₄", American Journal of Mathematics, 105 (2): 539–594, doi:10.2307/2374268, ISSN 0002-9327, JSTOR 2374268, MR 0701569
• Tits, Jacques (1989), "Groupes associés aux algèbres de Kac-Moody", Astérisque, Séminaire Bourbaki (177): 7–31, ISSN 0303-1179, MR 1040566
• Wilson, Robert A. (2010), "Another new approach to the small Ree groups", Archiv der Mathematik, 94 (6): 501–510, CiteSeerX 10.1.1.156.9909, doi:10.1007/s00013-010-0130-4, ISSN 0003-9268, MR 2653666, S2CID 122724281
• Wilson, Robert A. (2010b), "A simple construction of the Ree groups of type ²F₄", Journal of Algebra, 323 (5): 1468–1481, doi:10.1016/j.jalgebra.2009.11.015, ISSN 0021-8693, MR 2584965

External links

• ATLAS: Ree group R(27)