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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.
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.
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.
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