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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
- What is the order of the cyclic subgroup of U5 generated by a = cos + i sin ? 10 3 a b 108 degrees 10 darrow_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_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_forward
- 1.e. Consider the dihedral group (D3, ◦), ie. the symmetry group of an equilateral triangle. Determine whether (D3, ◦) has a subset of order 2.arrow_forwardAB 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_forwardLet Dn be the dihedral group of order 2n. Decide whether the following pairs of groups are isomorphic or not. In each case, explain why. (i) Ds and Q8 × Z2. (ii) D6 and S3 × Z2. (iii) D6 and A4. (iv) D6 x Z2 and S4.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
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