Create Plane Figures w/ Non-Isomorphic 12 Element Symmetry Groups

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SUMMARY

This discussion focuses on creating two plane figures that each possess a 12-element symmetry group, specifically one being the dihedral group D_6 and the other being the cyclic group Z_12. The regular hexagon represents D_6, while a 12-bladed windmill shape effectively demonstrates Z_12, as it lacks reflection symmetry. The key conclusion is that these two groups are not isomorphic due to their differing symmetry properties.

PREREQUISITES
  • Understanding of symmetry groups in geometry
  • Familiarity with dihedral groups and cyclic groups
  • Knowledge of isomorphism in group theory
  • Basic skills in drawing geometric figures
NEXT STEPS
  • Study the properties of dihedral groups, specifically D_6
  • Explore cyclic groups, focusing on Z_12 and its characteristics
  • Learn about group isomorphism and its implications in symmetry
  • Practice drawing various geometric figures to identify their symmetry groups
USEFUL FOR

Students of abstract algebra, geometry enthusiasts, and educators teaching group theory concepts related to symmetry in plane figures.

MikeDietrich
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Homework Statement


Draw two plane figures, each having a 12 element group of symmetries, such that the two groups are NOT isomorphic. Demonstrate that they are not isomorphic.

Homework Equations


I know that every finite group of isometries of the plane is isomorphic to either Z_n or to the dihedral group D_n.

The Attempt at a Solution


I drew a regular hexagon (D_6) but now I am stuck as to what to draw for a figure to represent Z_12. Would a 12 bladed windmill (pinwheel) type shape with pronged ends work?
 
Last edited:
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Yes, this will work, since it is not preserved by a reflection.
 

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