# Algebra group structures for dihedral group:D8

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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:

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