Reading Question 6.2.1. Let G be a cyclic group of order 20. Which of the following claims is true?Select all that apply.(a) G has 5 elements of order 4.(b) G has 4 elements of order 5.(c) G has 1 element of order 2.(d) G contains 10 generators.(e) If a is a generator of G, then the order of ak is equal to the gcd(20, k).

We know, if G is a cyclic group of order n and d | n, then number of elements in G of order d is…

Answered: Reading Question 6.2.1. Let G be a…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.8: Some Results On Finite Abelian Groups (optional)

Problem 15E: 15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that...

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15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .
27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.
Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.
9. Find all homomorphic images of the octic group.
For each of the following values of n, find all distinct generators of the group Un described in Exercise 11. a. n=7 b. n=5 c. n=11 d. n=13 e. n=17 f. n=19
Exercises 10. Find an isomorphism from the multiplicative group to the group with multiplication table in Figure . This group is known as the Klein four group. Figure Sec. 16. a. Prove that each of the following sets is a subgroup of , the general linear group of order over . Sec. 3. Let be the Klein four group with its multiplication table given in Figure . Figure Sec. 17. Show that a group of order either is cyclic or is isomorphic to the Klein four group . Sec. 16. Repeat Exercise with the quaternion group , the Klein four group , and defined by
Exercises 21. Suppose is a cyclic group of order. Determine the number of generators of for each value of and list all the distinct generators of . a. b. c. d. e. f.
Exercises 9. For each of the following values of, find all distinct generators of the cyclic group under addition. a. b. c. d. e. f.
17. Let and be elements of a group. Prove that is abelian if and only if .
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 .
Exercises In Section 3.3, the centralizer of an element a in the group G was shown to be the subgroup given by Ca=xGax=xa. Use the multiplication table constructed in Exercise 20 to find the centralizer Ca for each element a of the octic group D4. Construct a multiplication table for the octic group D4 described in Example 12 of this section.

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