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Which one of the following is True?
- a)Zn is cyclic if and only if n is prime.
- b)Every proper subgroup of Zn is cyclic.
- c)Every proper subgroup of S4 is cyclic.
- d)If every proper subgroup of a group is cyclic, then the group is cyclic.
Correct answer is option 'B'. Can you explain this answer?
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Which one of the following is True?a)Zn is cyclic if and only if n is ...
Explanation:
The given question is related to group theory. Let's understand the options one by one.
a) Zn is cyclic if and only if n is prime: This statement is not true. For example, Z4 is cyclic, but 4 is not a prime number.
b) Every proper subgroup of Zn is cyclic: This statement is true. Let's understand it with an example. Consider Z8 = {0, 1, 2, 3, 4, 5, 6, 7}. Now, let H be a proper subgroup of Z8. H can be {0}, {0, 4}, {0, 2, 4, 6}, or {0, 1, 2, 3, 4, 5, 6, 7}. We can easily verify that all these subgroups are cyclic.
c) Every proper subgroup of S4 is cyclic: This statement is not true. For example, the subgroup generated by (12)(34) in S4 is not cyclic.
d) If every proper subgroup of a group is cyclic, then the group is cyclic: This statement is not true. For example, the Klein four-group is a group in which every proper subgroup is cyclic, but the group itself is not cyclic.
Therefore, the correct answer is option B.
In Summary:
- Option A is false.
- Option B is true.
- Option C is false.
- Option D is false.
The given question is related to group theory. Let's understand the options one by one.
a) Zn is cyclic if and only if n is prime: This statement is not true. For example, Z4 is cyclic, but 4 is not a prime number.
b) Every proper subgroup of Zn is cyclic: This statement is true. Let's understand it with an example. Consider Z8 = {0, 1, 2, 3, 4, 5, 6, 7}. Now, let H be a proper subgroup of Z8. H can be {0}, {0, 4}, {0, 2, 4, 6}, or {0, 1, 2, 3, 4, 5, 6, 7}. We can easily verify that all these subgroups are cyclic.
c) Every proper subgroup of S4 is cyclic: This statement is not true. For example, the subgroup generated by (12)(34) in S4 is not cyclic.
d) If every proper subgroup of a group is cyclic, then the group is cyclic: This statement is not true. For example, the Klein four-group is a group in which every proper subgroup is cyclic, but the group itself is not cyclic.
Therefore, the correct answer is option B.
In Summary:
- Option A is false.
- Option B is true.
- Option C is false.
- Option D is false.
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Which one of the following is True?a)Zn is cyclic if and only if n is ...
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Which one of the following is True?a)Zn is cyclic if and only if n is prime.b)Every proper subgroup of Zn is cyclic.c)Every proper subgroup of S4 is cyclic.d)If every proper subgroup of a group is cyclic, then the group is cyclic.Correct answer is option 'B'. Can you explain this answer?
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Which one of the following is True?a)Zn is cyclic if and only if n is prime.b)Every proper subgroup of Zn is cyclic.c)Every proper subgroup of S4 is cyclic.d)If every proper subgroup of a group is cyclic, then the group is cyclic.Correct answer is option 'B'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Which one of the following is True?a)Zn is cyclic if and only if n is prime.b)Every proper subgroup of Zn is cyclic.c)Every proper subgroup of S4 is cyclic.d)If every proper subgroup of a group is cyclic, then the group is cyclic.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following is True?a)Zn is cyclic if and only if n is prime.b)Every proper subgroup of Zn is cyclic.c)Every proper subgroup of S4 is cyclic.d)If every proper subgroup of a group is cyclic, then the group is cyclic.Correct answer is option 'B'. Can you explain this answer?.
Which one of the following is True?a)Zn is cyclic if and only if n is prime.b)Every proper subgroup of Zn is cyclic.c)Every proper subgroup of S4 is cyclic.d)If every proper subgroup of a group is cyclic, then the group is cyclic.Correct answer is option 'B'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Which one of the following is True?a)Zn is cyclic if and only if n is prime.b)Every proper subgroup of Zn is cyclic.c)Every proper subgroup of S4 is cyclic.d)If every proper subgroup of a group is cyclic, then the group is cyclic.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which one of the following is True?a)Zn is cyclic if and only if n is prime.b)Every proper subgroup of Zn is cyclic.c)Every proper subgroup of S4 is cyclic.d)If every proper subgroup of a group is cyclic, then the group is cyclic.Correct answer is option 'B'. Can you explain this answer?.
Solutions for Which one of the following is True?a)Zn is cyclic if and only if n is prime.b)Every proper subgroup of Zn is cyclic.c)Every proper subgroup of S4 is cyclic.d)If every proper subgroup of a group is cyclic, then the group is cyclic.Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for IIT JAM.
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