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Let C^ * = C / \{0\} denote the group of non-zero complex numbers under multiplication. Suppose Y n =\ z in C| z ^ n = 1 \ . n \in N Which of the following is(are) subgroup(s) of C*?
(a) bigcup s = 1 ^ 100 Y s.
(b) bigcup n = 1 ^ (∞) Y 2^ *
(c) bigcup n = 100 ^ (∞) Y n
(d) bigcup n = 1 ^ (∞) Y n?
(a) bigcup s = 1 ^ 100 Y s.
(b) bigcup n = 1 ^ (∞) Y 2^ *
(c) bigcup n = 100 ^ (∞) Y n
(d) bigcup n = 1 ^ (∞) Y n?
Most Upvoted Answer
Let C^ * = C / \{0\} denote the group of non-zero complex numbers unde...
Understanding the Group C*
C* refers to the group of non-zero complex numbers under multiplication. The elements Y_n represent the nth roots of unity, defined as the complex numbers z such that z^n = 1.
Subgroup Criteria
To determine if a set is a subgroup of C*, it must satisfy:
- Contain the identity element (1).
- Be closed under the group operation (multiplication).
- Be closed under taking inverses.
Analysis of Each Option
- (a) bigcup s = 1 ^ 100 Y s
- This union includes all roots of unity from Y_1 to Y_100.
- Contains the identity (1): Yes, since Y_1 = {1}.
- Closed under multiplication: Yes, the product of any two roots of unity is another root of unity.
- Closed under inverses: Yes, each root has an inverse also in the set.
- Conclusion: This is a subgroup.
- (b) bigcup n = 1 ^ (∞) Y 2^*
- This union includes all 2^n roots of unity for n ≥ 1.
- Contains the identity (1): Yes.
- Closed under multiplication: Yes, since roots of unity multiplied together yield another root of unity.
- Closed under inverses: Yes.
- Conclusion: This is a subgroup.
- (c) bigcup n = 100 ^ (∞) Y n
- This union includes all roots of unity from Y_100 onward.
- Contains the identity (1): Yes.
- Closed under multiplication: Yes.
- Closed under inverses: Yes.
- Conclusion: This is a subgroup.
- (d) bigcup n = 1 ^ (∞) Y n
- This union includes all roots of unity from Y_1 onward.
- Contains the identity (1): Yes.
- Closed under multiplication: Yes.
- Closed under inverses: Yes.
- Conclusion: This is a subgroup.
Final Verdict
All options (a), (b), (c), and (d) are subgroups of C*.
C* refers to the group of non-zero complex numbers under multiplication. The elements Y_n represent the nth roots of unity, defined as the complex numbers z such that z^n = 1.
Subgroup Criteria
To determine if a set is a subgroup of C*, it must satisfy:
- Contain the identity element (1).
- Be closed under the group operation (multiplication).
- Be closed under taking inverses.
Analysis of Each Option
- (a) bigcup s = 1 ^ 100 Y s
- This union includes all roots of unity from Y_1 to Y_100.
- Contains the identity (1): Yes, since Y_1 = {1}.
- Closed under multiplication: Yes, the product of any two roots of unity is another root of unity.
- Closed under inverses: Yes, each root has an inverse also in the set.
- Conclusion: This is a subgroup.
- (b) bigcup n = 1 ^ (∞) Y 2^*
- This union includes all 2^n roots of unity for n ≥ 1.
- Contains the identity (1): Yes.
- Closed under multiplication: Yes, since roots of unity multiplied together yield another root of unity.
- Closed under inverses: Yes.
- Conclusion: This is a subgroup.
- (c) bigcup n = 100 ^ (∞) Y n
- This union includes all roots of unity from Y_100 onward.
- Contains the identity (1): Yes.
- Closed under multiplication: Yes.
- Closed under inverses: Yes.
- Conclusion: This is a subgroup.
- (d) bigcup n = 1 ^ (∞) Y n
- This union includes all roots of unity from Y_1 onward.
- Contains the identity (1): Yes.
- Closed under multiplication: Yes.
- Closed under inverses: Yes.
- Conclusion: This is a subgroup.
Final Verdict
All options (a), (b), (c), and (d) are subgroups of C*.
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Let C^ * = C / \{0\} denote the group of non-zero complex numbers under multiplication. Suppose Y n =\ z in C| z ^ n = 1 \ . n \in N Which of the following is(are) subgroup(s) of C*? (a) bigcup s = 1 ^ 100 Y s. (b) bigcup n = 1 ^ (∞) Y 2^ *(c) bigcup n = 100 ^ (∞) Y n(d) bigcup n = 1 ^ (∞) Y n?
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Let C^ * = C / \{0\} denote the group of non-zero complex numbers under multiplication. Suppose Y n =\ z in C| z ^ n = 1 \ . n \in N Which of the following is(are) subgroup(s) of C*? (a) bigcup s = 1 ^ 100 Y s. (b) bigcup n = 1 ^ (∞) Y 2^ *(c) bigcup n = 100 ^ (∞) Y n(d) bigcup n = 1 ^ (∞) Y n? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let C^ * = C / \{0\} denote the group of non-zero complex numbers under multiplication. Suppose Y n =\ z in C| z ^ n = 1 \ . n \in N Which of the following is(are) subgroup(s) of C*? (a) bigcup s = 1 ^ 100 Y s. (b) bigcup n = 1 ^ (∞) Y 2^ *(c) bigcup n = 100 ^ (∞) Y n(d) bigcup n = 1 ^ (∞) Y n? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let C^ * = C / \{0\} denote the group of non-zero complex numbers under multiplication. Suppose Y n =\ z in C| z ^ n = 1 \ . n \in N Which of the following is(are) subgroup(s) of C*? (a) bigcup s = 1 ^ 100 Y s. (b) bigcup n = 1 ^ (∞) Y 2^ *(c) bigcup n = 100 ^ (∞) Y n(d) bigcup n = 1 ^ (∞) Y n?.
Let C^ * = C / \{0\} denote the group of non-zero complex numbers under multiplication. Suppose Y n =\ z in C| z ^ n = 1 \ . n \in N Which of the following is(are) subgroup(s) of C*? (a) bigcup s = 1 ^ 100 Y s. (b) bigcup n = 1 ^ (∞) Y 2^ *(c) bigcup n = 100 ^ (∞) Y n(d) bigcup n = 1 ^ (∞) Y n? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let C^ * = C / \{0\} denote the group of non-zero complex numbers under multiplication. Suppose Y n =\ z in C| z ^ n = 1 \ . n \in N Which of the following is(are) subgroup(s) of C*? (a) bigcup s = 1 ^ 100 Y s. (b) bigcup n = 1 ^ (∞) Y 2^ *(c) bigcup n = 100 ^ (∞) Y n(d) bigcup n = 1 ^ (∞) Y n? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let C^ * = C / \{0\} denote the group of non-zero complex numbers under multiplication. Suppose Y n =\ z in C| z ^ n = 1 \ . n \in N Which of the following is(are) subgroup(s) of C*? (a) bigcup s = 1 ^ 100 Y s. (b) bigcup n = 1 ^ (∞) Y 2^ *(c) bigcup n = 100 ^ (∞) Y n(d) bigcup n = 1 ^ (∞) Y n?.
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