UPSC Exam > UPSC Questions > Let C be the field of complex numbers and C *...
Start Learning for Free
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?
(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.
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.
Attention UPSC Students!
To make sure you are not studying endlessly, EduRev has designed UPSC study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in UPSC.
|
Explore Courses for UPSC exam
|
|
Similar UPSC Doubts
Top Courses for UPSCView all
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?
Question Description
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?.
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?.
Solutions 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? in English & in Hindi are available as part of our courses for UPSC.
Download more important topics, notes, lectures and mock test series for UPSC Exam by signing up for free.
Here you can find the meaning of 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? defined & explained in the simplest way possible. Besides giving the explanation of
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?, a detailed solution 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? has been provided alongside types of 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? theory, EduRev gives you an
ample number of questions to practice 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? tests, examples and also practice UPSC tests.
|
|
Explore Courses for UPSC exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup