Answer to Question #144058 in Abstract Algebra for Ms

Question

https://www.chegg.com/homework-help/questions-and-answers/d4-find-elements-x-x-3-v-x-3-r90-x-3-r0-x-2-r0-x-2-h-dy-dihedral-group-order-8-dy-r0-r90-r-q14939929

Answer 1

$$Solution$$

Given D₄ is the dihedral group of order 8

a. ) X³ = V

The only solution to this equation is X = V

$$\therefore V^3 = V \cdot V^2= V \cdot R_0 = V$$

Note that; for square of any element here, see the main diagonal.

b. ) X³ = R₉₀

The only solution to this is X = R₂₇₀

$$\therefore R^3_{270} = R_{270} \cdot R^2_{270} = R_{270} \cdot R_{180} = R_{90}$$

c. ) X³ = R₀

No element in dihedral group has order 3.

So, the only solution to this is X = R₀

d. ) X² = R₀

See the main diagonal.

Solutions are,

$$X=R_0, R_{180},H,V,D, \ and\ D'$$

e. ) X² = H

Again, see the main diagonal. There is no H there.

So, this equation has no solution.