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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?
(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-ze...
Understanding the Properties of C and C^*
The field of complex numbers, denoted as C, and the group of non-zero complex numbers under multiplication, denoted as C^*, have unique properties. Let's analyze the given statements.
(a) C is cyclic
- C is not cyclic because it is an infinite-dimensional vector space over the real numbers. A cyclic group can be generated by a single element, which is not possible for the entire field of complex numbers. Thus, this statement is false.
(b) Every finite subgroup of C^* is cyclic
- Every finite subgroup of C^* is indeed cyclic. This is a consequence of the structure theorem for finite abelian groups, where finite subgroups of C^* can be expressed as roots of unity. Hence, this statement is true.
(c) C^* has finitely many finite subgroups
- C^* contains infinitely many finite subgroups, as it includes all nth roots of unity for any integer n. Each set of nth roots of unity forms a distinct finite subgroup. Therefore, this statement is false.
(d) Every proper subgroup of C^* is cyclic
- This statement is not universally true. While all finite subgroups are cyclic, infinite proper subgroups may not be. For instance, the subgroup of all non-zero complex numbers with a fixed modulus (which is infinite) is not cyclic. Thus, this statement is false.
Summary of Truth Values
- (a) False
- (b) True
- (c) False
- (d) False
In conclusion, only statement (b) holds true in the context of complex numbers and their multiplicative group.
The field of complex numbers, denoted as C, and the group of non-zero complex numbers under multiplication, denoted as C^*, have unique properties. Let's analyze the given statements.
(a) C is cyclic
- C is not cyclic because it is an infinite-dimensional vector space over the real numbers. A cyclic group can be generated by a single element, which is not possible for the entire field of complex numbers. Thus, this statement is false.
(b) Every finite subgroup of C^* is cyclic
- Every finite subgroup of C^* is indeed cyclic. This is a consequence of the structure theorem for finite abelian groups, where finite subgroups of C^* can be expressed as roots of unity. Hence, this statement is true.
(c) C^* has finitely many finite subgroups
- C^* contains infinitely many finite subgroups, as it includes all nth roots of unity for any integer n. Each set of nth roots of unity forms a distinct finite subgroup. Therefore, this statement is false.
(d) Every proper subgroup of C^* is cyclic
- This statement is not universally true. While all finite subgroups are cyclic, infinite proper subgroups may not be. For instance, the subgroup of all non-zero complex numbers with a fixed modulus (which is infinite) is not cyclic. Thus, this statement is false.
Summary of Truth Values
- (a) False
- (b) True
- (c) False
- (d) False
In conclusion, only statement (b) holds true in the context of complex numbers and their multiplicative group.
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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?.
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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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
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