Exercise 15. all have a modulus of 1; that is The set T is defined as the subset of C whose elem T = {c € C : |e| = 1} (a) Using Proposition 15.4.6 above, prove that T is a subgroup of C*. (b) What is |T|? (c) Prove or disprove that T is abelian.
Exercise 15. all have a modulus of 1; that is The set T is defined as the subset of C whose elem T = {c € C : |e| = 1} (a) Using Proposition 15.4.6 above, prove that T is a subgroup of C*. (b) What is |T|? (c) Prove or disprove that T is abelian.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.4: Cosets Of A Subgroup
Problem 32E: (See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup...
Related questions
Question
Please do Exercise 15.4.9 part ABCD and please show step by step and explain
![Proposition 15.4.6. A subset H of a group G is a subgroup if and only if:
(a) The identity e of G is in H.
(b) If h1, h2 € H, then hịh2 e H (that is, H is closed under the group
operation),
(c) If h e H, then h-l e H.
Exercise 15.4.7. The set T is defined as the subset of C whose elements
all have a modulus of 1; that is
T = {c € C : |c| = 1}
(a) Using Proposition 15.4.6 above, prove that T is a subgroup of C*.
(b) What is |T|?
(c) Prove or disprove that T is abelian.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F76e8ac28-349b-4e70-bf3f-6def7ab9b1c8%2Feyqenil_processed.png&w=3840&q=75)
Transcribed Image Text:Proposition 15.4.6. A subset H of a group G is a subgroup if and only if:
(a) The identity e of G is in H.
(b) If h1, h2 € H, then hịh2 e H (that is, H is closed under the group
operation),
(c) If h e H, then h-l e H.
Exercise 15.4.7. The set T is defined as the subset of C whose elements
all have a modulus of 1; that is
T = {c € C : |c| = 1}
(a) Using Proposition 15.4.6 above, prove that T is a subgroup of C*.
(b) What is |T|?
(c) Prove or disprove that T is abelian.
![Exercise 15.4.9. Let's generalize the last exercise. Suppose now that Hn
is the set of nth roots of unity. That is
H, = {z € C : 2" = 1}
(a) Prove that Hn is a subset of T.
(b) Using Proposition 15.4.6, prove that H is a subgroup of T.
(c) What is |Hn|?
(d) Prove or disprove that Hn is abelian.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F76e8ac28-349b-4e70-bf3f-6def7ab9b1c8%2Ft2465vu_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 15.4.9. Let's generalize the last exercise. Suppose now that Hn
is the set of nth roots of unity. That is
H, = {z € C : 2" = 1}
(a) Prove that Hn is a subset of T.
(b) Using Proposition 15.4.6, prove that H is a subgroup of T.
(c) What is |Hn|?
(d) Prove or disprove that Hn is abelian.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,