Algebra group structures for dihedral group:D8

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:

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