3D4

{{Short description|Family of mathematical groups}} {{DISPLAYTITLE:³D₄}}In mathematics, the Steinberg triality groups of type '''³D₄''' form a family of Steinberg or twisted Chevalley groups. They are quasi-split forms of D₄, depending on a cubic Galois extension of fields ''K'' ⊂ ''L'', and using the triality automorphism of the Dynkin diagram D₄. Unfortunately the notation for the group is not standardized, as some authors write it as ³D₄(''K'') (thinking of ³D₄ as an algebraic group taking values in ''K'') and some as ³D₄(''L'') (thinking of the group as a subgroup of D₄(''L'') fixed by an outer automorphism of order 3). The group ³D₄ is very similar to an orthogonal or spin group in dimension 8.

Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced by {{harvtxt|Steinberg|1959}}. They were independently discovered by Jacques Tits in {{harvtxt|Tits|1958}} and {{harvtxt|Tits|1959}}.

==Construction==

The simply connected split algebraic group of type D₄ has a triality automorphism σ of order 3 coming from an order 3 automorphism of its Dynkin diagram. If ''L'' is a field with an automorphism τ of order 3, then this induced an order 3 automorphism τ of the group D₄(''L''). The group ³D₄(''L'') is the subgroup of D₄(''L'') of points fixed by στ. It has three 8-dimensional representations over the field ''L'', permuted by the outer automorphism τ of order 3.

==Over finite fields==

The group ³D₄(''q''³) has order ''q''¹² (''q''⁸ + ''q''⁴ + 1) (''q''⁶ − 1) (''q''² − 1). For comparison, the split spin group D₄(''q'') in dimension 8 has order ''q''¹² (''q''⁸ − 2''q''⁴ + 1) (''q''⁶ − 1) (''q''² − 1) and the quasisplit spin group ²D₄(''q''²) in dimension 8 has order ''q''¹² (''q''⁸ − 1) (''q''⁶ − 1) (''q''² − 1).

The group ³D₄(''q''³) is always simple. The Schur multiplier is always trivial. The outer automorphism group is cyclic of order ''f'' where ''q''³ = ''p^f'' and ''p'' is prime.

This group is also sometimes called ³''D''₄(''q''), ''D''₄²(''q''³), or a twisted Chevalley group.

==³D₄(2³)==

The smallest member of this family of groups has several exceptional properties not shared by other members of the family. It has order 211341312 = 2¹²⋅3⁴⋅7²⋅13 and outer automorphism group of order 3.

The automorphism group of ³D₄(2³) is a maximal subgroup of the Thompson sporadic group, and is also a subgroup of the compact Lie group of type F₄ of dimension 52. In particular it acts on the 26-dimensional representation of F₄. In this representation it fixes a 26-dimensional lattice that is the unique 26-dimensional even lattice of determinant 3 with no norm 2 vectors, studied by {{harvtxt|Elkies|Gross|1996}}. The dual of this lattice has 819 pairs of vectors of norm 8/3, on which ³D₄(2³) acts as a rank 4 permutation group.

The group ³D₄(2³) has 9 classes of maximal subgroups, of structure : 2<SUP>1+8</SUP>:L<SUB>2</SUB>(8) fixing a point of the rank 4 permutation representation on 819 points. : [2<SUP>11</SUP>]:(7 × S<SUB>3</SUB>) : U<SUB>3</SUB>(3):2 : S<SUB>3</SUB> × L<SUB>2</SUB>(8) : (7 × L<SUB>2</SUB>(7)):2 : 3<SUP>1+2</SUP>.2S<SUB>4</SUB> : 7<SUP>2</SUP>:2A<SUB>4</SUB> : 3<SUP>2</SUP>:2A<SUB>4</SUB> : 13:4

==See also==

*List of finite simple groups *²E₆

==References==

*{{Citation | last1=Carter | first1=Roger W. | author1-link=Roger Carter (mathematician) | title=Simple groups of Lie type | orig-year=1972 | publisher=John Wiley & Sons | location=New York | series=Wiley Classics Library | isbn=978-0-471-50683-6 | mr=0407163 | year=1989}} *{{Citation | last1=Elkies | first1=Noam D. | last2=Gross | first2=Benedict H. | title=The exceptional cone and the Leech lattice | doi=10.1155/S1073792896000426 | mr=1411589 | year=1996 | journal=International Mathematics Research Notices | volume=1996 | issn=1073-7928 | issue=14 | pages=665–698 | doi-access=free }} *{{Citation | last1=Steinberg | first1=Robert | title=Variations on a theme of Chevalley | url=https://projecteuclid.org/euclid.pjm/1103039126 | mr=0109191 | year=1959 | journal=Pacific Journal of Mathematics | 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=https://www.math.ucla.edu/~rst/ |publisher=Yale University, New Haven, Conn. |mr=0466335 |year=1968 |url-status=dead |archiveurl=https://web.archive.org/web/20120910032654/http://www.math.ucla.edu/~rst/ |archivedate=2012-09-10 }} *{{Citation | last1=Tits | first1=Jacques | title=Les "formes réelles" des groupes de type E₆ | url=https://www.numdam.org/item?id=SB_1956-1958__4__351_0 | publisher=Secrétariat math'ematique | location=Paris | series=Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152 à 168; 2e èd. corrigée, Exposé 162 | mr=0106247 | year=1958 | volume=15}} *{{Citation | last1=Tits | first1=Jacques | title=Sur la trialité et certains groupes qui s'en déduisent | journal = Inst. Hautes Études Sci. Publ. Math. | year=1959 | volume=2 | pages = 13–60| doi=10.1007/BF02684706 | s2cid=120426125 | url=https://www.numdam.org/item/PMIHES_1959__2__13_0/ }}

==External links== *[https://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TD42/ ³D₄(2³) at the atlas of finite groups] *[https://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TD43/ ³D₄(3³) at the atlas of finite groups]

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Category:Finite groups Category:Lie groups