UCLA Abstract Algebra Actions on The Dihedral Group & Orbit Stabilizer Theorem Exercise

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Requisite information is provided in the attached image.Requirements: Everything must be written out clearly and all theorems utilized MUST be cited.

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We are going to use the standard action to make a different action of D12 on a set
with five elements. Consider the picture
5
6
4
1
3
2
Let A = {1,3,5} be the vertices of the red triangle, let B = {2,4,6} be the vertices
of the blue triangle, and let X = {1,4}, Y = {2,5), and Z = {3,6} be the vertices
of the three green diagonals. The action of D12 on the set of vertices {1,2,3,4,5,6}
induces an action of D12 on the set {A, B, X, Y, Z}.
(A) Compute r· A, r.B, r.X, r. Y, and r. Z. Do the same with r replaced
by s.
(B) What is the orbit of A? What is the orbit of X?
(C) What is the stabilizer of X? Hint: you can find the order of the stabilizer
quickly using the orbit stabilizer theorem.
(D) Write down the permutations 0, and o, in Ss corresponding to the
elements res e D12.

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Tags:
Permutations

Abstract Algebra

Algebra Actions

The Dihedral Group

Orbit Stabilizer Theorem

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