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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 cyclic?
(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 cyclic?
Most Upvoted Answer
Let C be the field of complex numbers and C* be the group of non-zero ...
Analysis of C*
The group C* of non-zero complex numbers under multiplication has interesting properties. Let’s evaluate the statements one by one.
(a) C* is cyclic
- C* is not cyclic. A cyclic group can be generated by a single element; however, C* contains elements of infinite order and cannot be generated by one element. For example, any element with an argument (angle) that is not a rational multiple of π cannot generate all of C*.
(b) Every finite subgroup of C* is cyclic
- This statement is true. Any finite subgroup of C* can be represented by roots of unity, which are all powers of a single complex number (the primitive root of unity). Hence, finite subgroups must be cyclic.
(c) C* has finitely many finite subgroups
- This statement is false. C* has infinitely many finite subgroups, as for each integer n, there exists a subgroup generated by the n-th roots of unity. Thus, there are infinitely many finite subgroups.
(d) Every proper subgroup of C* is cyclic
- This statement is true. Proper subgroups of C* consist of either finite subgroups or infinite subgroups that are not the whole group. Any proper subgroup of C* that is finite is cyclic, while infinite proper subgroups (like those generated by elements of the form re^(iθ) where θ is irrational) are also cyclic since they can be generated by a single element.
Conclusion
- True statements: (b) and (d)
- False statements: (a) and (c)
The group C* of non-zero complex numbers under multiplication has interesting properties. Let’s evaluate the statements one by one.
(a) C* is cyclic
- C* is not cyclic. A cyclic group can be generated by a single element; however, C* contains elements of infinite order and cannot be generated by one element. For example, any element with an argument (angle) that is not a rational multiple of π cannot generate all of C*.
(b) Every finite subgroup of C* is cyclic
- This statement is true. Any finite subgroup of C* can be represented by roots of unity, which are all powers of a single complex number (the primitive root of unity). Hence, finite subgroups must be cyclic.
(c) C* has finitely many finite subgroups
- This statement is false. C* has infinitely many finite subgroups, as for each integer n, there exists a subgroup generated by the n-th roots of unity. Thus, there are infinitely many finite subgroups.
(d) Every proper subgroup of C* is cyclic
- This statement is true. Proper subgroups of C* consist of either finite subgroups or infinite subgroups that are not the whole group. Any proper subgroup of C* that is finite is cyclic, while infinite proper subgroups (like those generated by elements of the form re^(iθ) where θ is irrational) are also cyclic since they can be generated by a single element.
Conclusion
- True statements: (b) and (d)
- False statements: (a) and (c)
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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 cyclic?
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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 cyclic? 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 cyclic? 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 cyclic?.
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 cyclic? 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 cyclic? 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 cyclic?.
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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 cyclic?, 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 cyclic? 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 cyclic? theory, EduRev gives you an
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