Group Actions on Truncated Octahedron

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SUMMARY

The discussion focuses on the group of rotational symmetries of the octahedron, denoted as G, and its action on the edges of the truncated octahedron. Participants explore the orbits formed by this action, concluding that the edges of the square faces and hexagonal faces create distinct orbits. A key point raised is the concept of representative elements within these orbits, with emphasis on identifying stabilizers for chosen edges. The consensus leans towards a transitive group action, particularly noting the stabilizer's role in determining orbit sizes.

PREREQUISITES
  • Understanding of group theory, specifically group actions
  • Familiarity with rotational symmetries of polyhedra
  • Knowledge of orbits and stabilizers in group actions
  • Basic visualization skills for geometric shapes, particularly polyhedra
NEXT STEPS
  • Study the properties of group actions in abstract algebra
  • Learn about the rotational symmetries of the octahedron and their implications
  • Investigate the concept of orbits and stabilizers in greater detail
  • Explore visual tools or software for modeling polyhedral symmetries
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in geometric group theory will benefit from this discussion, particularly those studying symmetries of polyhedra.

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Homework Statement


Let G be the group of rotational symmetries of the octahedron and consider the action of G on the edges of the truncated octahedron.

Describe the orbits of this action.

Choose one representative element in each orbit. Describe the stabilizers of these representative elements.

Homework Equations





The Attempt at a Solution


I am having a lot of trouble visualizing the movements of the edges of the truncated octahedron (which is what I am assuming is what happens when G acts on them). But I think the edges of the square faces make up one orbit and I'm not sure on the others. Any thoughts?

Also, I don't really know what it means to pick a representative element of an orbit. Just pick one edge and see what the stabilizers are?
 
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well if you pick an edge and see what the stabilisers are, this can tell you the size of the orbit, at least that way you know what you're looking for.

when i was looking at this my gut feeling was that the edges of the square faces would be one orbit and those of the hexagonal faces another but after some thought, the edges of the square faces are also edges of the hexagonal faces. I'm going to go for transitive group action for this reason and the fact that using the symmetries of the octahedron i can't find anything other than e to stabilise a particular edge.
 
Maybe this doesn't make sense, but I don't see how the edges of the square can be in the same orbit as the edges that only border hexagons.
 

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