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Problem 2.55. Consider the dihedral group D3 introduced in Problem 2.21. To give us a common...

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Advance Math

Problem 2.55. Consider the dihedral group D3 introduced inProblem 2.21. To give us a common starting point, let’s assume thetriangle and hole are positioned so that one of the tips of thetriangle is pointed up. Let r be rotation by 120◦ in the clockwisedirection and let s be the reflection in D3 that fixes the top ofthe triangle.

(a) Describe the action of r −1 on the triangle and express r −1as a word using r only.

(b) Describe the action of s −1 on the triangle and express s −1as a word using s only.
(c) Prove that D3 = hr, si by writing every element of D3 as a wordin r or s.

(d) Is {r, s} a minimal generating set for D3 ?

(e) Explain why there is no single generating set for D3consisting of a single element. This proves that D3 is notcyclic.

It is important to point out that the fact that {r, s} is aminimal generating set for D3 does not imply that D3 is not acyclic group. There are examples of cyclic groups that have minimalgenerating sets consisting of more than one element (see Problem2.70).

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