Exercises
Construct a multiplication table for the octic group
Example
Using the notational convention described in the preceding paragraph, we shall write out the dihedral group
The elements of the group
1. the identity mapping
2. the counterclockwise rotation
3. the counterclockwise rotation
4. the counterclockwise rotation
5. the reflection
6. the reflection
7. the reflection
8. the reflection
The dihedral group
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Chapter 4 Solutions
Elements Of Modern Algebra
- Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.arrow_forward45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )arrow_forward9. Find all homomorphic images of the octic group.arrow_forward
- 1. Construct the multiplication table for the the group Us = {1,a, a², a°, aª} 2ni where a = e 5. 2. Give all the elements of Ug and write them in rectangular form. Draw the regular octagon in the complex plane whose vertices are the elements of Ug. 3. Use Cardano's method to solve the equation x3 in rectangular form. 6x – 9 = 0. Write the rootsarrow_forwardDetermine the symmetry group in the following figures. For cyclic group (Cn), determine the order of rotation; and for dihedral groups (Dn), determine the number of reflection lines.arrow_forwardWhat is the order of the cyclic subgroup of U5 generated by a = cos + i sin ? 10 3 a b 108 degrees 10 darrow_forward
- AB is rotated 120° clockwise about B. Then AB is rotated 45° counterclockwise about A. What is the image of A as a composition of transformations? A.(r(120°, B) ∘ r(–45°, A))(A) B.(r(–45°, A) ∘ r(120°, B))(A) C.(r(–120°, B) ∘ r(45°, A))(A) D.(r(45°, A) ∘ r(–120°, B))(A)arrow_forwardWrite out the Cayley table for the group {1, -1, i, -i} under multiplication where i = sqrt(-1). (In general, we apply the row element on the left of the column element.)arrow_forwardI know the symmetry group of this figure is supposed to be of size 8, but all I can think of are the identity and three rotations of 90 degrees each. Any idea of what the other four motions would be? Thanks.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning