UCLA Abstract Algebra Transitive and Stabilizer of D12 Set of Vertices Exercise

Description

Part 1: Prove that the action is transitivePart 2: Compute the stabilizer of the vertex
Both parts MUST be written out rigorously with all theorems used cited.

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In this question you will think about D12. Remember that we have the standard
presentation
D12 = (r, s|96 = 52 = 1, rs = sr=1)
Consider the usual action of D12 on the regular hexagon, as pictured below:
5
6
4
1
3
2
As we discussed in lecture, we can think of this as an action of D12 on the set of vertices
{1,2,3,4,5,6}. Specifically, in terms of the induced permutation representation, the
rotation r corresponds to the permuation (123456) ES6, and the reflection s corresponds
to (26)(35) E $6.

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Tags:
Abstract Algebra

HEXAGON

Figure vertices

Vertex stabilizer

Transitive action

permutation representation

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