1. Construct the multiplication table for the the group Us = {1,a, a“, a°, a*} %3D 2mi where a = es.
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- In Exercises 3 and 4, let be the octic group in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let be the subgroup of the octic group . Find the distinct left cosets of in , write out their elements, partition into left cosets of , and give . Find the distinct right cosets of in , write out their elements, and partition into right cosets of . Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group of rigid motions of a square The elements of the group are as follows: 1. the identity mapping 2. the counterclockwise rotation through about the center 3. the counterclockwise rotation through about the center 4. the counterclockwise rotation through about the center 5. the reflection about the horizontal line 6. the reflection about the diagonal 7. the reflection about the vertical line 8. the reflection about the diagonal . The dihedral group of rigid motions of the square is also known as the octic group. The multiplication table for is requested in Exercise 20 of this section.In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group D4 of rigid motions of a square The elements of the group D4 are as follows: 1. the identity mapping e=(1) 2. the counterclockwise rotation =(1,2,3,4) through 900 about the center O 3. the counterclockwise rotation 2=(1,3)(2,4) through 1800 about the center O 4. the counterclockwise rotation 3=(1,4,3,2) through 2700 about the center O 5. the reflection =(1,4)(2,3) about the horizontal line h 6. the reflection =(2,4) about the diagonal d1 7. the reflection =(1,2)(3,4) about the vertical line v 8. the reflection =(1,3) about the diagonal d2. The dihedral group D4=e,,2,3,,,, of rigid motions of the square is also known as the octic group. The multiplication table for D4 is requested in Exercise 20 of this section.Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .
- Let a,b,c, and d be elements of a group G. Find an expression for (abcd)1 in terms of a1,b1,c1, and d1.For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication. (Sec. 3.5,3,6, Sec. 4.6,17). Find an isomorphism from the additive group 4={ [ 0 ]4,[ 1 ]4,[ 2 ]4,[ 3 ]4 } to the multiplicative group of units U5={ [ 1 ]5,[ 2 ]5,[ 3 ]5,[ 4 ]5 }5. Find an isomorphism from the additive group 6={ [ a ]6 } to the multiplicative group of units U7={ [ a ]77[ a ]7[ 0 ]7 }. Repeat Exercise 14 where G is the multiplicative group of units U20 and G is the cyclic group of order 4. That is, G={ [ 1 ],[ 3 ],[ 7 ],[ 9 ],[ 11 ],[ 13 ],[ 17 ],[ 19 ] }, G= a =e,a,a2,a3 Define :GG by ([ 1 ])=([ 11 ])=e ([ 3 ])=([ 13 ])=a ([ 9 ])=([ 19 ])=a2 ([ 7 ])=([ 17 ])=a3.Find all subgroups of the octic group D4.