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

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

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

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

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

(e) Explain why there is no single generating set for D3 consisting of a single element. This proves that D3 is not cyclic.

It is important to point out that the fact that {r, s} is a minimal generating set for D3 does not imply that D3 is not a cyclic group. There are examples of cyclic groups that have minimal generating sets consisting of more than one element (see Problem 2.70).

Expert Solution

Colby Messinger answered 4 months ago

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