1. Let p e Z be a prime number and set Z, = { e Q : If ged(n, m) = 1, then p {m}.%3Dd. Show that every proper subgroup of Z(p®) is finite. Hint: Show that if U < Z(p®) is infinite, thenU = Z(p®).

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Answered: 1. Let p e Z be a prime number and set…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 21E: With H and K as in Exercise 18, prove that K is a normal subgroup of HK. Exercise18: If H is a...

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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.

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1. Let p e Z be a prime number and set Z, = {" e Q : If ged(n, m) = 1, then p {m}. %3D d. Show that every proper subgroup of Z(p®) is finite. Hint: Show that if U