Algebraic group interpretations of dihedral group:D8

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This article gives specific information, namely, algebraic group interpretations, about a particular group, namely: 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)