Misplaced Pages

Ree group: Difference between revisions

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Browse history interactively← Previous editContent deleted Content addedVisualWikitext
Revision as of 01:04, 7 May 2014 editQfmd (talk | contribs)5 edits Afd: Nominated for deletion; see Misplaced Pages:Articles for deletion/Ree_group← Previous edit Latest revision as of 00:02, 3 December 2023 edit undoOAbot (talk | contribs)Bots441,761 editsm Open access bot: doi updated in citation with #oabot. 
(37 intermediate revisions by 25 users not shown)
Line 1: Line 1:
<!-- Please do not remove or change this AfD message until the issue is settled -->
{{Article for deletion/dated|page=Ree group|timestamp=20140507010446|year=2014|month=May|day=7|substed=yes}}
<!-- For administrator use only: {{Old AfD multi|page=Ree group|date=7 May 2014|result='''keep'''}} -->
<!-- End of AfD message, feel free to edit beyond this point -->

In mathematics, a '''Ree group''' is a ] over a ] constructed by {{harvs|txt|last=Ree|year1=1960|year2=1961|authorlink=Rimhak Ree}} from an exceptional ] of a ] that reverses the direction of the multiple bonds, generalizing the ] found by Suzuki using a different method. They were the last of the infinite families of ]s to be discovered. In mathematics, a '''Ree group''' is a ] over a ] constructed by {{harvs|txt|last=Ree|year1=1960|year2=1961|authorlink=Rimhak Ree}} from an exceptional ] of a ] that reverses the direction of the multiple bonds, generalizing the ] found by Suzuki using a different method. They were the last of the infinite families of ]s to be discovered.


Unlike the ]s, the Ree groups are not given by the points of a connected ] 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 ]s 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. Unlike the ]s, the Ree groups are not given by the points of a connected ] 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 ]s 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.


{{harvtxt|Tits|1960}} defined Ree groups over infinite fields of characteristics 2 and 3. {{harvtxt|Tits|1989}} and {{harvtxt|Hée|1990}} introduced Ree groups of infinite-dimensional ]s. {{harvtxt|Tits|1960}} defined Ree groups over infinite fields of characteristics 2 and 3. {{harvtxt|Tits|1989}} and {{harvtxt|Hée|1990}} introduced Ree groups of infinite-dimensional ]s.
Line 12: Line 7:
==Construction== ==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: If {{mvar|X}} is a ], Chevalley constructed split algebraic groups corresponding to {{mvar|X}}, in particular giving groups {{math|''X''(''F'')}} with values in a field {{mvar|F}}. These groups have the following automorphisms:
*Any endomorphism σ of the field ''F'' induces an endomorphism α<sub>σ</sub> of the group ''X''(''F'') *Any endomorphism {{mvar|σ}} of the field {{mvar|F}} induces an endomorphism {{mvar|α<sub>σ</sub>}} of the group {{math|''X''(''F'')}}
*Any automorphism π of the Dynkin diagram induces an automorphism α<sub>π</sub> of the group ''X''(''F''). *Any automorphism {{mvar|π}} of the Dynkin diagram induces an automorphism {{mvar|α<sub>π</sub>}} of the group {{math|''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. The Steinberg and Chevalley groups can be constructed as fixed points of an endomorphism of ''X''(''F'') for {{mvar|F}} the algebraic closure of a field. For the Chevalley groups, the automorphism is the Frobenius endomorphism of {{mvar|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<sub>φ</sub> associated to the Frobenius endomorphism φ of the field ''F''. Roughly speaking, this endomorphism α<sub>π</sub> comes from the order 2 automorphism of the Dynkin diagram where one ignores the lengths of the roots. Over fields of characteristic 2 the groups {{math|B<sub>2</sub>(''F'')}} and {{math|F<sub>4</sub>(''F'')}} and over fields of characteristic 3 the groups {{math|G<sub>2</sub>(''F'')}} have an endomorphism whose square is the endomorphism {{mvar|α<sub>φ</sub>}} associated to the Frobenius endomorphism {{mvar|φ}} of the field {{mvar|F}}. Roughly speaking, this endomorphism {{mvar|α<sub>π</sub>}} 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<sub>π</sub>(''g'') = α<sub>σ</sub>(''g''). If the field ''F'' is perfect then α<sub>π</sub> and α<sub>φ</sub> are automorphisms, and the Ree group is the group of fixed points of the involution α<sub>φ</sub>/α<sub>π</sub> of ''X''(''F''). Suppose that the field {{mvar|F}} has an endomorphism {{mvar|σ}} whose square is the Frobenius endomorphism: {{math|''σ''<sup>2</sup> {{=}} ''φ''}}. Then the Ree group is defined to be the group of elements {{mvar|g}} of {{math|''X''(''F'')}} such that {{math|''α<sub>π</sub>''(''g'') {{=}} ''α<sub>σ</sub>''(''g'')}}. If the field {{mvar|F}} is perfect then {{mvar|α<sub>π</sub>}} and {{mvar|α<sub>φ</sub>}} are automorphisms, and the Ree group is the group of fixed points of the involution {{mvar|α<sub>φ</sub>/α<sub>π</sub>}} of {{math|''X''(''F'')}}.


In the case when ''F'' is a finite field of order ''p''<sup>''k''</sup> (with ''p'' = 2 or 3) there is an endomorphism with square the Frobenius exactly when ''k'' = 2''n'' + 1 is odd, in which case it is unique. In the case when {{mvar|F}} is a finite field of order {{mvar|p<sup>k</sup>}} (with ''p'' = 2 or 3) there is an endomorphism with square the Frobenius exactly when ''k'' = 2''n'' + 1 is odd, in which case it is unique. So this gives the finite Ree groups as subgroups of B<sub>2</sub>(2<sup>2''n''+1</sup>), F<sub>4</sub>(2<sup>2''n''+1</sup>), and G<sub>2</sub>(3<sup>2''n''+1</sup>) fixed by an involution.
So this gives the finite Ree groups as subgroups of B₂(2<sup>2''n''+1</sup>), F₄(2<sup>2''n''+1</sup>), and G₂(3<sup>2''n''+1</sup>) fixed by an involution.


==Chevalley groups, Steinberg group, and Ree groups== ==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''({{overline|''F''}}) where {{overline|''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: 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 {{math|'''Z'''}} whose values over finite fields are the Chevalley groups. In general one can take the fixed points of an endomorphism {{mvar|α}} of {{math|''X''({{overline|''F''}})}} where {{math|{{overline|''F''}}}} is the algebraic closure of a finite field, such that some power of {{mvar|α}} is some power of the Frobenius endomorphism φ. The three cases are as follows:
*For Chevalley groups, α = φ<sup>''n''</sup> 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 Chevalley groups, {{math|''α'' {{=}} ''φ<sup>n</sup>''}} 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<sup>''m''</sup> = φ<sup>''n''</sup> 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 Steinberg groups, {{math|''α<sup>m</sup>'' {{=}} ''φ<sup>n</sup>''}} 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<sup>''m''</sup> = φ<sup>''n''</sup> 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 ''p''<sup>''n''</sup> with ''n'' odd, and there is no corresponding field of order ''p''<sup>''n''/2</sup> (although some authors like to pretend there is in their notation for the groups). *For Ree groups, {{math|''α<sup>m</sup>'' {{=}} ''φ<sup>n</sup>''}} 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 {{mvar|p<sup>n</sup>}} with ''n'' odd, and there is no corresponding field of order ''p''<sup>''n''/2</sup> (although some authors like to pretend there is in their notation for the groups).


==Ree groups of type <sup>2</sup>B<sub>2</sub>== ==Ree groups of type <sup>2</sup>B<sub>2</sub>==
Line 36: Line 30:
{{main|Suzuki groups}} {{main|Suzuki groups}}


The Ree groups of type ²B₂ were first found by {{harvtxt|Suzuki|1960}} using a different method, and are usually called ]. Ree noticed that they could be constructed from the groups of type ''B''<sub>2</sub> using a variation of the construction of {{harvtxt|Steinberg|1959}}. Ree realized that a similar construction could be applied to the Dynkin diagrams ''F''<sub>4</sub> and ''G''<sub>2</sub>, leading to two new families of finite simple groups. The Ree groups of type <sup>2</sup>B<sub>2</sub> were first found by {{harvtxt|Suzuki|1960}} using a different method, and are usually called ]. Ree noticed that they could be constructed from the groups of type B<sub>2</sub> using a variation of the construction of {{harvtxt|Steinberg|1959}}. Ree realized that a similar construction could be applied to the Dynkin diagrams F<sub>4</sub> and G<sub>2</sub>, leading to two new families of finite simple groups.


==Ree groups of type <sup>2</sup>G<sub>2</sub>== ==Ree groups of type <sup>2</sup>G<sub>2</sub>==


The Ree groups of type <sup>2</sup>G<sub>2</sub>(3<sup>2''n''+1</sup>) were introduced by {{harvtxt|Ree|1960}}, who showed that they are all simple except for the first one <sup>2</sup>G<sub>2</sub>(3), which is isomorphic to the automorphism group of SL<sub>2</sub>(8). {{harvtxt|Wilson|2010}} gave a simplified construction of the Ree groups, as the automorphisms of a 7-dimensional vector space over the field with 3<sup>2''n''+1</sup> elements preserving a bilinear form, a trilinear form, and a bilinear product. The Ree groups of type <sup>2</sup>G<sub>2</sub>(3<sup>2''n''+1</sup>) were introduced by {{harvtxt|Ree|1960}}, who showed that they are all simple except for the first one <sup>2</sup>G<sub>2</sub>(3), which is isomorphic to the automorphism group of {{math|SL<sub>2</sub>(8)}}. {{harvtxt|Wilson|2010}} gave a simplified construction of the Ree groups, as the automorphisms of a 7-dimensional vector space over the field with 3<sup>2''n''+1</sup> elements preserving a bilinear form, a trilinear form, and a product satisfying a twisted linearity law.


The Ree group has order {{math|''q''<sup>3</sup>(''q''<sup>3</sup> + 1)(''q'' − 1)}} where ''q'' = 3<sup>2''n''+1</sup>
The Ree group has order
''q''<sup>3</sup>
(''q''<sup>3</sup>&nbsp;+&nbsp;1)
(''q''&nbsp;−&nbsp;1)
where
''q'' = 3<sup>2''n''+1</sup>


The Schur multiplier is trivial for ''n''&nbsp;≥&nbsp;1 and for <sup>2</sup>''G''<sub>2</sub>(3)′. The Schur multiplier is trivial for ''n''&nbsp;≥&nbsp;1 and for <sup>2</sup>''G''<sub>2</sub>(3)′.
Line 53: Line 42:
The outer automorphism group is cyclic of order&nbsp;2''n''&nbsp;+&nbsp;1. The outer automorphism group is cyclic of order&nbsp;2''n''&nbsp;+&nbsp;1.


The Ree group is also occasionally denoted by Ree(''q''), R(''q''), or E<sub>2</sub><sup>*</sup>(''q'') The Ree group is also occasionally denoted by Ree(''q''), R(''q''), or E<sub>2</sub><sup>*</sup>(''q'')


The Ree group <sup>2</sup>''G''<sub>2</sub>(''q'') has a ] on ''q''<sup>3</sup>&nbsp;+&nbsp;1 points, and more precisely acts as automorphisms of an S(2, ''q''+1, ''q''<sup>3</sup>+1) ]. It also acts on a 7-dimensional vector space over the field with ''q'' elements as it is a subgroup of ''G''<sub>2</sub>(''q''). The Ree group <sup>2</sup>G<sub>2</sub>(''q'') has a ] on {{math|''q''<sup>3</sup> + 1}} points, and more precisely acts as automorphisms of an S(2, ''q''+1, ''q''<sup>3</sup>+1) ]. It also acts on a 7-dimensional vector space over the field with ''q'' elements as it is a subgroup of G<sub>2</sub>(''q'').


The 2-sylow subgroups of the Ree groups are elementary abelian of order 8. ] 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 ]. 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'''&nbsp;&times;&nbsp;PSL<sub>2</sub>(''q''), and by investigating groups with an involution centralizer of the similar form '''Z'''/2'''Z'''&nbsp;&times;&nbsp;PSL<sub>2</sub>(5) Janko found the sporadic group&nbsp;]. {{harvtxt|Kleidman|1988}} determined their maximal subgroups. The 2-sylow subgroups of the Ree groups are elementary abelian of order 8. ] 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 ]. These groups also played a role in the discovery of the first modern sporadic group. They have involution centralizers of the form {{math|'''Z'''/2'''Z''' × PSL<sub>2</sub>(''q'')}}, and by investigating groups with an involution centralizer of the similar form {{math|'''Z'''/2'''Z''' × PSL<sub>2</sub>(5)}} Janko found the sporadic group&nbsp;]. {{harvtxt|Kleidman|1988}} determined their maximal subgroups.


The Ree groups of type <sup>2</sup>G<sub>2</sub> are exceptionally hard to characterize. {{harvs|txt|last=Thompson|year1=1967|year2=1972|year3=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 {{harvs|txt|last=Bombieri|year=1980}} used ] to show that Thompson's conditions implied that σ<sup>2</sup>=3 in all but 178 small cases, that were eliminated using a computer by ] and Hunt. Bombieri found out about this problem after reading an article about the classification by {{harvtxt|Gorenstein|1979}},who suggested that someone from outside group theory might be able to help solving it. {{harvtxt|Enguehard|1986}} gave a unified account of the solution of this problem by Thompson and Bombieri. The Ree groups of type <sup>2</sup>G<sub>2</sub> are exceptionally hard to characterize. {{harvs|txt|last=Thompson|year1=1967|year2=1972|year3=1977}} studied this problem, and was able to show that the structure of such a group is determined by a certain automorphism {{mvar|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 {{mvar|σ}}. Finally {{harvs|txt|last=Bombieri|year=1980}} used ] to show that Thompson's conditions implied that {{math|''σ''<sup>2</sup> {{=}} 3}} in all but 178 small cases, that were eliminated using a computer by ] and Hunt. Bombieri found out about this problem after reading an article about the classification by {{harvtxt|Gorenstein|1979}}, who suggested that someone from outside group theory might be able to help solving it. {{harvtxt|Enguehard|1986}} gave a unified account of the solution of this problem by Thompson and Bombieri.


==Ree groups of type <sup>2</sup>F<sub>4</sub>== ==Ree groups of type <sup>2</sup>F<sub>4</sub>==


The Ree groups of type <sup>2</sup>F<sub>4</sub>(2<sup>2''n''+1</sup>) were introduced by {{harvtxt|Ree|1961}}. They are simple except for the first one <sup>2</sup>F<sub>4</sub>(2), which {{harvtxt|Tits|1964}} showed has a simple subgroup of index 2, now known as the ]. {{harvtxt|Wilson|2010b}} gave a simplified construction of the Ree groups as the symmetries of a 26-dimensional space over the field of order 2<sup>2''n''+1</sup> preserving a quadratic form, a cubic form, and a partial multiplication. The Ree groups of type {{math|<sup>2</sup>F<sub>4</sub>(2<sup>2''n''+1</sup>)}} were introduced by {{harvtxt|Ree|1961}}. They are simple except for the first one {{math|<sup>2</sup>F<sub>4</sub>(2)}}, which {{harvtxt|Tits|1964}} showed has a simple subgroup of index 2, now known as the ].
{{harvtxt|Wilson|2010b}} gave a simplified construction of the Ree groups as the symmetries of a 26-dimensional space over the field of order 2<sup>2''n''+1</sup> preserving a quadratic form, a cubic form, and a partial multiplication.


The Ree group <sup>2</sup>F<sub>4</sub>(2<sup>2''n''+1</sup>) has order The Ree group {{math|<sup>2</sup>F<sub>4</sub>(2<sup>2''n''+1</sup>)}} has order
''q''<sup>12</sup> ''q''<sup>12</sup>(''q''<sup>6</sup>&nbsp;+&nbsp;1)
(''q''<sup>6</sup>&nbsp;+&nbsp;1)
(''q''<sup>4</sup>&nbsp;−&nbsp;1) (''q''<sup>4</sup>&nbsp;−&nbsp;1)
(''q''<sup>3</sup>&nbsp;+&nbsp;1) (''q''<sup>3</sup>&nbsp;+&nbsp;1)
Line 76: Line 65:
The ] is cyclic of order&nbsp;2''n''&nbsp;+&nbsp;1. The ] is cyclic of order&nbsp;2''n''&nbsp;+&nbsp;1.


These Ree groups have the unusual property that the ] of their ] is not crystallographic: it is the dihedral group of order 16. {{harvtxt|Tits|1983}} showed that all ]s come from Ree groups of type&nbsp;²F₄. These Ree groups have the unusual property that the ] of their ] is not crystallographic: it is the dihedral group of order 16. {{harvtxt|Tits|1983}} showed that all ]s come from Ree groups of type {{math|<sup>2</sup>F<sub>4</sub>}}.


==See also== ==See also==
{{portal|Mathematics}}

*] *]


==References== ==References==


*{{Citation | last1=Carter | first1=Roger W. | author1-link=Roger Carter (mathematician) | title=Simple groups of Lie type | origyear=1972 | publisher=] | location=New York | series=Wiley Classics Library | isbn=978-0-471-50683-6 | mr=0407163 | year=1989}} *{{Citation | last1=Carter | first1=Roger W. | author1-link=Roger Carter (mathematician) | title=Simple groups of Lie type | orig-year=1972 | publisher=] | location=New York | series=Wiley Classics Library | isbn=978-0-471-50683-6 | mr=0407163 | year=1989}}
*{{Citation | last1=Bombieri | first1=Enrico | author1-link=Enrico Bombieri | others=appendices by Andrew Odlyzko and D. Hunt | title=Thompson's problem (σ²=3) | url=http://dx.doi.org/10.1007/BF01402275 | doi=10.1007/BF01402275 | id={{MR|570875}} | year=1980 | journal=] | issn=0020-9910 | volume=58 | issue=1 | pages=77–100}} *{{Citation | last1=Bombieri | first1=Enrico | author1-link=Enrico Bombieri | others=appendices by Andrew Odlyzko and D. Hunt | title=Thompson's problem (σ<sup>2</sup>=3) | doi=10.1007/BF01402275 |mr=570875 | year=1980 | journal=] | issn=0020-9910 | volume=58 | issue=1 | pages=77–100| s2cid=122867511 }}
*{{Citation | last1=Enguehard | first1=Michel | title=Caractérisation des groupes de Ree | id={{MR|873958}} | year=1986 | journal=Astérisque | issn=0303-1179 | issue=142 | pages=49–139}} *{{Citation | last1=Enguehard | first1=Michel | title=Caractérisation des groupes de Ree |mr=873958 | year=1986 | journal=Astérisque | issn=0303-1179 | issue=142 | pages=49–139}}
*{{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=The classification of finite simple groups. I. Simple groups and local analysis | doi=10.1090/S0273-0979-1979-14551-8 | year=1979 | journal=American Mathematical Society. Bulletin. New Series | issn=0002-9904 | volume=1 | issue=1 | pages=43–199 | mr=513750}} *{{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=The classification of finite simple groups. I. Simple groups and local analysis | doi=10.1090/S0273-0979-1979-14551-8 | year=1979 | journal=Bulletin of the American Mathematical Society |series=New Series | issn=0002-9904 | volume=1 | issue=1 | pages=43–199 | mr=513750| doi-access=free }}
*{{Citation | last1=Hée | first1=Jean-Yves | title=Construction de groupes tordus en théorie de Kac-Moody | id={{MR|1044619}} | year=1990 | journal=Comptes Rendus de l'Académie des Sciences. Série I. Mathématique | issn=0764-4442 | volume=310 | issue=3 | pages=77–80}} *{{Citation | last1=Hée | first1=Jean-Yves | title=Construction de groupes tordus en théorie de Kac-Moody |mr=1044619 | year=1990 | journal=Comptes Rendus de l'Académie des Sciences, Série I | issn=0764-4442 | volume=310 | issue=3 | pages=77–80}}
*{{Citation | last1=Kleidman | first1=Peter B. | title=The maximal subgroups of the Chevalley groups G₂(q) with q odd, the Ree groups ²G₂(q), and their automorphism groups | url=http://dx.doi.org/10.1016/0021-8693(88)90239-6 | doi=10.1016/0021-8693(88)90239-6 | id={{MR|955589}} | year=1988 | journal=] | issn=0021-8693 | volume=117 | issue=1 | pages=30–71}} *{{Citation | last1=Kleidman | first1=Peter B. | title=The maximal subgroups of the Chevalley groups G<sub>2</sub>(q) with q odd, the Ree groups <sup>2</sup>G<sub>2</sub>(q), and their automorphism groups | doi=10.1016/0021-8693(88)90239-6 |mr=955589 | year=1988 | journal=] | issn=0021-8693 | volume=117 | issue=1 | pages=30–71| doi-access= }}
*{{Citation | last1=Ree | first1=Rimhak | title=A family of simple groups associated with the simple Lie algebra of type (G<sub>2</sub>) | url=http://www.ams.org/journals/bull/1960-66-06/S0002-9904-1960-10523-X/home.html | doi=10.1090/S0002-9904-1960-10523-X | mr=0125155 | year=1960 | journal=] | issn=0002-9904 | volume=66 | pages=508–510}} *{{Citation | last1=Ree | first1=Rimhak | title=A family of simple groups associated with the simple Lie algebra of type (G<sub>2</sub>) | url=https://www.ams.org/journals/bull/1960-66-06/S0002-9904-1960-10523-X/home.html | doi=10.1090/S0002-9904-1960-10523-X | mr=0125155 | year=1960 | journal=] | issn=0002-9904 | volume=66 | issue=6 | pages=508–510| doi-access=free }}
*{{Citation | last1=Ree | first1=Rimhak | title=A family of simple groups associated with the simple Lie algebra of type (F<sub>4</sub>) | url=http://www.ams.org/journals/bull/1961-67-01/S0002-9904-1961-10527-2/home.html | doi=10.1090/S0002-9904-1961-10527-2 | mr=0125155 | year=1961 | journal=] | issn=0002-9904 | volume=67 | pages=115–116}} *{{Citation | last1=Ree | first1=Rimhak | title=A family of simple groups associated with the simple Lie algebra of type (F<sub>4</sub>) | url=https://www.ams.org/journals/bull/1961-67-01/S0002-9904-1961-10527-2/home.html | doi=10.1090/S0002-9904-1961-10527-2 | mr=0125155 | year=1961 | journal=] | issn=0002-9904 | volume=67 | pages=115–116| doi-access=free }}
*{{Citation | last1=Steinberg | first1=Robert | title=Variations on a theme of Chevalley | url=http://projecteuclid.org/euclid.pjm/1103039126 | mr=0109191 | year=1959 | journal=] | issn=0030-8730 | volume=9 | pages=875–891}} *{{Citation | last1=Steinberg | first1=Robert | title=Variations on a theme of Chevalley | url=http://projecteuclid.org/euclid.pjm/1103039126 | mr=0109191 | year=1959 | journal=] | issn=0030-8730 | volume=9 | issue=3 | pages=875–891 | doi=10.2140/pjm.1959.9.875| doi-access=free }}
*{{Citation | last1=Steinberg | first1=Robert | title=Lectures on Chevalley groups | url=http://www.math.ucla.edu/~rst/ | publisher=Yale University, New Haven, Conn. | mr=0466335 | year=1968}} *{{Citation | last1=Steinberg | first1=Robert | title=Lectures on Chevalley groups | url=https://www.math.ucla.edu/~rst/ | publisher=Yale University, New Haven, Conn. | mr=0466335 | year=1968 | url-status=dead | archive-url=https://web.archive.org/web/20120910032654/http://www.math.ucla.edu/~rst/ | archive-date=2012-09-10 }}
*{{Citation | last1=Steinberg | first1=Robert | title=Endomorphisms of linear algebraic groups | url=http://books.google.com/books?id=54HO1wDNM_YC | publisher=] | location=Providence, R.I. | series=Memoirs of the American Mathematical Society, No. 80 | id={{MR|0230728}} | year=1968}} *{{Citation | last1=Steinberg | first1=Robert | title=Endomorphisms of linear algebraic groups | url=https://books.google.com/books?id=54HO1wDNM_YC | publisher=] | location=Providence, R.I. | series=Memoirs of the American Mathematical Society, No. 80 |mr=0230728 | year=1968| isbn=9780821812808 }}
*{{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki | title=A new type of simple groups of finite order | jstor=70960 | mr=0120283 | year=1960 | journal=] | issn=0027-8424 | volume=46 | pages=868–870}} *{{Citation | last1=Suzuki | first1=Michio | author1-link=Michio Suzuki (mathematician) | title=A new type of simple groups of finite order | jstor=70960 | mr=0120283 | year=1960 | journal=] | issn=0027-8424 | volume=46 | issue=6 | pages=868–870 | doi=10.1073/pnas.46.6.868 | pmid=16590684 | pmc=222949| doi-access=free }}
*{{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Toward a characterization of E<sub>2</sub>*(q) | doi=10.1016/0021-8693(67)90080-4 | id={{MR|0223448}} | year=1967 | journal=] | issn=0021-8693 | volume=7 | pages=406–414}} *{{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Toward a characterization of E<sub>2</sub>*(q) | doi=10.1016/0021-8693(67)90080-4 |mr=0223448 | year=1967 | journal=] | issn=0021-8693 | volume=7 | issue=3 | pages=406–414| doi-access= }}
*{{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Toward a characterization of E<sub>2</sub>*(q) . II | doi=10.1016/0021-8693(72)90074-9 | id={{MR|0313377}} | year=1972 | journal=] | issn=0021-8693 | volume=20 | pages=610–621}} *{{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Toward a characterization of E<sub>2</sub>*(q) . II | doi=10.1016/0021-8693(72)90074-9 |mr=0313377 | year=1972 | journal=] | issn=0021-8693 | volume=20 | issue=3 | pages=610–621| doi-access=free }}
*{{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Toward a characterization ofE<sub>2</sub>*(q) . III | doi=10.1016/0021-8693(77)90276-9 | id={{MR|0453858}} | year=1977 | journal=] | issn=0021-8693 | volume=49 | issue=1 | pages=162–166}} *{{Citation | last1=Thompson | first1=John G. | author1-link=John G. Thompson | title=Toward a characterization of E<sub>2</sub>*(q) . III | doi=10.1016/0021-8693(77)90276-9 |mr=0453858 | year=1977 | journal=] | issn=0021-8693 | volume=49 | issue=1 | pages=162–166| doi-access= }}
*{{Citation | last1=Tits | first1=Jacques | title=Séminaire Bourbaki, Vol. 6 | url=http://www.numdam.org/item?id=SB_1960-1961__6__65_0 | publisher=] | location=Paris | id={{MR|1611778}} | year=1960 | chapter=Les groupes simples de Suzuki et de Ree | pages=65–82}} *{{Citation | last1=Tits | first1=Jacques | title=Séminaire Bourbaki, Vol. 6 | chapter-url=http://www.numdam.org/item?id=SB_1960-1961__6__65_0 | publisher=] | location=Paris |mr=1611778 | year=1960 | chapter=Les groupes simples de Suzuki et de Ree | pages=65–82}}
*{{Citation | last1=Tits | first1=Jacques | title=Algebraic and abstract simple groups | jstor=1970394 | mr=0164968 | year=1964 | journal=] | issn=0003-486X | volume=80 | pages=313–329}} *{{Citation | last1=Tits | first1=Jacques | title=Algebraic and abstract simple groups | jstor=1970394 | mr=0164968 | year=1964 | journal=] |series=Second Series | issn=0003-486X | volume=80 | issue=2 | pages=313–329 | doi=10.2307/1970394}}
*{{Citation | last1=Tits | first1=Jacques | title=Moufang octagons and the Ree groups of type ²F₄ | url=http://dx.doi.org/10.2307/2374268 | doi=10.2307/2374268 | id={{MR|701569}} | year=1983 | journal=] | issn=0002-9327 | volume=105 | issue=2 | pages=539–594}} *{{Citation | last1=Tits | first1=Jacques | title=Moufang octagons and the Ree groups of type <sup>2</sup>F<sub>4</sub> | doi=10.2307/2374268 |mr=701569 | year=1983 | journal=] | issn=0002-9327 | volume=105 | issue=2 | pages=539–594| jstor=2374268 }}
*{{Citation | last1=Tits | first1=Jacques | title=Groupes associés aux algèbres de Kac-Moody | url=http://www.numdam.org/item?id=SB_1988-1989__31__7_0 | series=Séminaire Bourbaki | id={{MR|1040566}} | year=1989 | journal=Astérisque | issn=0303-1179 | issue=177 | pages=7–31}} *{{Citation | last1=Tits | first1=Jacques | title=Groupes associés aux algèbres de Kac-Moody | url=http://www.numdam.org/item?id=SB_1988-1989__31__7_0 | series=Séminaire Bourbaki |mr=1040566 | year=1989 | journal=Astérisque | issn=0303-1179 | issue=177 | pages=7–31}}
*{{Citation | last1=Wilson | first1=Robert A. | title=Another new approach to the small Ree groups | url=http://dx.doi.org/10.1007/s00013-010-0130-4 | doi=10.1007/s00013-010-0130-4 | id={{MR|2653666}} | year=2010 | journal=Archiv der Mathematik | issn=0003-9268 | volume=94 | issue=6 | pages=501–510}} *{{Citation | last1=Wilson | first1=Robert A. | title=Another new approach to the small Ree groups | doi=10.1007/s00013-010-0130-4 |mr=2653666 | year=2010 | journal=Archiv der Mathematik | issn=0003-9268 | volume=94 | issue=6 | pages=501–510| citeseerx=10.1.1.156.9909 | s2cid=122724281 }}
*{{Citation | last1=Wilson | first1=Robert A. | title=A simple construction of the Ree groups of type ²F₄ | url=http://dx.doi.org/10.1016/j.jalgebra.2009.11.015 | doi=10.1016/j.jalgebra.2009.11.015 | id={{MR|2584965}} | year=2010b | journal=] | issn=0021-8693 | volume=323 | issue=5 | pages=1468–1481}} *{{Citation | last1=Wilson | first1=Robert A. | title=A simple construction of the Ree groups of type <sup>2</sup>F<sub>4</sub> | doi=10.1016/j.jalgebra.2009.11.015 |mr=2584965 | year=2010b | journal=] | issn=0021-8693 | volume=323 | issue=5 | pages=1468–1481| doi-access= }}


==External links== ==External links==

* *


Latest revision as of 00:02, 3 December 2023

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 ασ of the group X(F)
  • Any automorphism π of the Dynkin diagram induces an automorphism απ 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 αφ associated to the Frobenius endomorphism φ of the field F. Roughly speaking, this endomorphism απ 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 = φ. Then the Ree group is defined to be the group of elements g of X(F) such that απ(g) = ασ(g). If the field F is perfect then απ and αφ are automorphisms, and the Ree group is the group of fixed points of the involution αφ/απ of X(F).

In the case when F is a finite field of order p (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(2), F4(2), and G2(3) 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 = φ 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 = φ 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 = φ 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 p with n odd, and there is no corresponding field of order p (although some authors like to pretend there is in their notation for the groups).

Ree groups of type B2

Main article: Suzuki groups

The Ree groups of type B2 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 G2

The Ree groups of type G2(3) were introduced by Ree (1960), who showed that they are all simple except for the first one G2(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 3 elements preserving a bilinear form, a trilinear form, and a product satisfying a twisted linearity law.

The Ree group has order q(q + 1)(q − 1) where q = 3

The Schur multiplier is trivial for n ≥ 1 and for G2(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 G2(q) has a doubly transitive permutation representation on q + 1 points, and more precisely acts as automorphisms of an S(2, q+1, q+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 G2 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 σ = 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 F4

The Ree groups of type F4(2) were introduced by Ree (1961). They are simple except for the first one F4(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 2 preserving a quadratic form, a cubic form, and a partial multiplication.

The Ree group F4(2) has order q(q + 1) (q − 1) (q + 1) (q − 1) where q = 2. 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 F4.

See also

References

External links

Category: