a) is a G = Z24 cyclic? Why b) Find all subgroups of G = Z24 %3D c) Let G = Z24 ,H =< 6 > find |G: H| d) Find the elements of Z24/<8> 8 > e) Find U6(18) Z24/<8> f) What is the order of the element 16+< 8 > in the factor group g) how we can Classification of Groups of Order 2p h) Explain The Converse of Lagrange's Theorem Is False CS CamScanner - igo ä>gusaall

a) is a G = Z24 cyclic? Why b) Find all subgroups of G = Z24 %3D c) Let G = Z24 ,H =< 6 > find |G: H| d) Find the elements of Z24/<8> 8 > e) Find U6(18) Z24/<8> f) What is the order of the element 16+< 8 > in the factor group g) how we can Classification of Groups of Order 2p h) Explain The Converse of Lagrange's Theorem Is False CS CamScanner - igo ä>gusaall

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 19E

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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.
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