Group Action
Consider D3 in the geometrical way, as symmetry group of isometries of the plane, and consider the corresponding group action on a set of 30 evenly spaced points on a circle, numbered 0 to 29, with 0 at one of the reflexion axes.
This section illustrates group action concepts for this case.
The action of G on X is called
- transitive if for any two x, y in X there exists an g in G such that g·x = y; - this is not the case
- faithful (or effective) if for any two different g, h in G there exists an x in X such that g·x ≠ h·x; - this is the case, because, except for the identity, symmetry groups do not contain elements that "do nothing"
- free if for any two different g, h in G and all x in X we have g·x ≠ h·x; - this is not the case because there are reflections
Read more about this topic: Dihedral Group Of Order 6
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