Question

ANSWER 2 & 3 please. Show work for my understanding and upvote. THANK YOU!!

2. Given a regular n-gon, let r be a rotation of it by 2π/n radians. This time, assume that we are not allowed to flip over the n-gon. These n actions form a group denotecd (a) Draw a Cayley diagram for Cn for n-4, n-5, and n-6 (b) For n 4, 5, 6, find all minimal generating sets of C.· [Note: There are minimal generating sets of of size 2.] (c) Make a conjecture of what integers k does C,-〈rk〉 for a general fixed integer n ix svinne tries of an equilateral triangle 쇼 form a group 3. As we saw in lecture, the s denoted Ds -{e,r, r2,f,rf,r^f], where r is a 120 clockwise rotation and f is a flip about a vertical axis (which fixes the top corner). Since and f suffice to generate all six of these symmetries, we write D,-(nf) (a) Let g be the reflection of the triangle that fixes the lower-left corner. Which of the six actions in D3 is g equal to? Which action is fg? (b) Write all 6 actions of D3 using only f and g. Draw a Cayley diagram using f and g as generators (c) To generate D3, we need at least 2 actions. It is not difficult to show that if we have 3 generators, then one of them is unnecessary. Find all minimal generating sets of Ds e,r, r2,f,rf,r2 f); note that all of them should have exactly two actions. Do not use qa in this list The triangle puzzle Let's play our "rectangle puzzle" game but with an equilaterial triangle: The "triangle puzzle" group, often denoted D3, has 6 actions 2 3 The identity action:e A (clockwise) 120° rotation: r A (clockwise) 240° rotation: r2 A horizontal flip: f 2 Rotate + horizontal flip: rf Rotate twice horizontal flip: rf. 2 1 2 One set of generators: D -(r,f) Notice that multiple paths can lead us to the same node. These give us relations in our group. For example This group is non-abelian: rf fr

Answer 1

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