3 Dimensional Representation of D3

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SUMMARY

The discussion focuses on obtaining the three-dimensional representation of the Dihedral group of order 6, denoted as D_3. The group elements are defined as D_3 = {e, c, c², b, bc, bc²}, where 'c' represents a 120° rotation about the z-axis and 'b' denotes reflections across the x-axis. Participants clarify that D_3 is isomorphic to S_3, which acts on ℝ³ by permuting standard basis vectors, but challenges arise in establishing a clear correspondence between the defined rotations and reflections in the context of three-dimensional representation.

PREREQUISITES
  • Understanding of group theory, specifically dihedral groups and their properties.
  • Familiarity with three-dimensional geometry and transformations in ℝ³.
  • Knowledge of linear algebra concepts, including matrix representations of transformations.
  • Basic understanding of isomorphism in group theory, particularly between D_3 and S_3.
NEXT STEPS
  • Study the matrix representation of the Dihedral group D_3 in three dimensions.
  • Explore the properties of the symmetric group S_3 and its action on ℝ³.
  • Learn about the geometric interpretation of group actions in three-dimensional space.
  • Investigate the relationship between rotations and reflections in the context of linear transformations.
USEFUL FOR

This discussion is beneficial for mathematicians, particularly those specializing in group theory, as well as students and researchers interested in geometric representations of algebraic structures.

ChrisVer
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I was wondering how can I obtain the three dimensional representation of the Dihedral group of order 6, [itex]D_3[/itex].

If this group has the elements: [itex]D_3 = \left \{ e,c,c^2,b,bc,bc^2 \right \}[/itex]

Where [itex]c[/itex] corresponds to rotation by [itex]120^o[/itex] on the xy plane (so about z-axis) and [itex]b[/itex] to reflections of the [itex]x[/itex] axis, I don't see how [itex]z[/itex] would change at all... so when I try to obtain it I'm getting the known 2-dimensional representation of [itex]D_3[/itex] together with an extra [itex]1[/itex] on the diagonal corresponding to the transformations [itex]z \rightarrow z'=z[/itex]...
Any help?
 
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[itex]D_3[/itex] is isomorphic to [itex]S_3[/itex], which acts naturally (but reducibly) on [itex]\mathbb{R}^3[/itex] by permuting the standard basis vectors.
 
So you are suggesting it will have the 3dim repr of S3?
I also thought that... but then the rotations and reflections as defined don't seem to help in this correspondence...:s
 

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