Algebra group structures for dihedral group:D8

From Groupprops
Jump to: navigation, search
This article gives specific information, namely, algebra group structures, about a particular group, namely: dihedral group:D8.
View algebra group structures for particular groups | View other specific information about dihedral group:D8

The group dihedral group:D8 has at least one (and probably only one?) algebra group structure over field:F2. It does not have any algebra group structure over any other fields.

Algebra

Multiplication table (structure constants)

The algebra is a three-dimensional algebra. We can describe it by means of the following multiplication table in terms of structure constants u,v,w. The multiplication table is as follows:

u v w
u 0 w 0
v 0 0 0
w 0 0 0

Verification of properties

  • The algebra is associative: All products of length three or more are zero.
  • The algebra is nilpotent: All products of length three or more are zero.
  • The algebra group is isomorphic to dihedral group:D8: 1 + w is the central element of order 2, 1 + u + v is the order four generator of the cyclic maximal subgroup. 1 + u and 1 + v are reflections outside this subgroup.

Description as subalgebra of niltriangular matrix Lie algebra

The algebra is the whole of niltriangular matrix Lie algebra:NT(3,2), so dihedral group:D8 is isomorphic to UT(3,2). The explicit description is:

u = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\\end{pmatrix}, \qquad v = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\\end{pmatrix}, \qquad w = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\\end{pmatrix}