3D4
In mathematics, ³D₄ is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form 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 Steinberg (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(q³) has order q12 (q8+q4+1) (q6−1) (q2−1). For comparison, the split spin group D₄(q) in dimension 8 has order q12 (q8–2q4+1) (q6−1) (q2−1) and the quasisplit spin group ²D₄(q²) in dimension 8 has order q12 (q8–1) (q6−1) (q2−1).
The group ³D₄(q³) is always simple. The Schur multiplier is always trivial. The outer automorphism group is cyclic of order f where q3 = pf and p is prime.
This group is also sometimes called 3D4(q), D42(q3), 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.
3D4(23) acts on the unique even 26 dimensional lattice of determinant 3 with no roots.
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.
It has 9 classes of maximal subgroups, of structure
21+8:L2(8) giving a rank 4 permutation representation on 819 points.
[211]:(7 × S3)
U3(3):2
S3 × L2(8)
(7 × L2(7)):2
31+2.2S4
72:2A4
32:2A4
13:4
See also
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
- Steinberg, Robert (1959), "Variations on a theme of Chevalley", Pacific Journal of Mathematics, 9: 875–891, ISSN 0030-8730, MR 0109191
- Steinberg, Robert (1968), Lectures on Chevalley groups, Yale University, New Haven, Conn., MR 0466335