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Let C be the field of complex numbers and C * be the group of non-zero complex numbers under multiplication. Then which of the following are true?
(a) C* is cyclic
(b) Every finite subgroup of C* is cyclic
(c) C* has finitely many finite subgroups
(d) Every proper subgroup of C* is eyelic?
Most Upvoted Answer
Let C be the field of complex numbers and C * be the group of non-zero...
Complex Numbers and Their Group Structure
The field of complex numbers, denoted as C, includes all numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit. The group of non-zero complex numbers, C*, forms a group under multiplication.
Analysis of Options
- (a) C* is cyclic
- False: A cyclic group is generated by a single element. C* is not cyclic because it contains numbers of various magnitudes and angles, and there is no single complex number that can represent all others through multiplication.
- (b) Every finite subgroup of C* is cyclic
- True: Finite subgroups of C* can be shown to be cyclic. Any finite subgroup of C* must consist of roots of unity, which can be generated by a single element (a primitive root of unity).
- (c) C* has finitely many finite subgroups
- False: C* has infinitely many finite subgroups. For example, for each integer n, the group of n-th roots of unity forms a distinct finite subgroup.
- (d) Every proper subgroup of C* is cyclic
- True: A proper subgroup of C* must be either finite or infinite. Any finite proper subgroup is cyclic, and any infinite proper subgroup of C* can be shown to be generated by a single element, thus also cyclic.
Conclusion
Based on the analysis:
- (a) False
- (b) True
- (c) False
- (d) True
Understanding the structure of C* helps in exploring deeper algebraic properties and group theory.
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Let C be the field of complex numbers and C * be the group of non-zero complex numbers under multiplication. Then which of the following are true?(a) C* is cyclic(b) Every finite subgroup of C* is cyclic(c) C* has finitely many finite subgroups(d) Every proper subgroup of C* is eyelic?
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Let C be the field of complex numbers and C * be the group of non-zero complex numbers under multiplication. Then which of the following are true?(a) C* is cyclic(b) Every finite subgroup of C* is cyclic(c) C* has finitely many finite subgroups(d) Every proper subgroup of C* is eyelic? 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 be the field of complex numbers and C * be the group of non-zero complex numbers under multiplication. Then which of the following are true?(a) C* is cyclic(b) Every finite subgroup of C* is cyclic(c) C* has finitely many finite subgroups(d) Every proper subgroup of C* is eyelic? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let C be the field of complex numbers and C * be the group of non-zero complex numbers under multiplication. Then which of the following are true?(a) C* is cyclic(b) Every finite subgroup of C* is cyclic(c) C* has finitely many finite subgroups(d) Every proper subgroup of C* is eyelic?.
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