Algebraic group interpretations of dihedral group:D8

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Unitriangular matrix group interpretation

Description

The group dihedral group:D8 can be interpreted as the group of $\mathbb{F}_2$-points of a unipotent algebraic group over the algebraic closure of $\mathbb{F}_2$, namely the unitriangular matrix group of degree three $UT(3,\overline{\mathbb{F}_2})$. Explicitly, $D_8 \cong UT(3,\mathbb{F}_2)$, which in turn is the set of $\mathbb{F}_2$-fixed points of $\overline{\mathbb{F}_2}$.

This is the only possible structure for $D_8$ as the $\mathbb{F}_2$-fixed points of a unipotent algebraic group.

In fact, since the group scheme corresponding to the unitriangular matrix group of degree three is integral (i.e., makes sense over $\mathbb{Z}$), analogues of the group can be defined over any ring.

Field extensions

Field extension of field:F2 Degree of extension Size of field Size of group = cube of size of field Group of points over that field in $UT(3,\overline{\mathbb{F}_2})$
field:F4 2 4 64 unitriangular matrix group:UT(3,4)
field:F8 3 8 512 unitriangular matrix group:UT(3,8)

Local rings with this as residue field

Local ring Length of ring Size of ring Size of group = cube of size of ring Corresponding algebraic group type notion
ring:Z4 2 4 64 unitriangular matrix group:UT(3,Z4) $\mathbb{F}_2[t]/(t^2)$ 2 4 64 unitriangular matrix group of degree three over quotient of polynomial ring over F2 by square of indeterminate
ring:Z8 3 8 512 unitriangular matrix group:UT(3,Z8)