IIT JAM Exam > IIT JAM Questions > Let G be a nonabelian group. Let G have ord...
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Let G be a nonabelian group. Let α ∈ G have order 4 and let β ∈ G have order 3. Then the order of the element αβ in G
- a)is 6
- b)is 12
- c)is of the form 12k for k ≥ 2
- d)need not be finite
Correct answer is option 'D'. Can you explain this answer?
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Let G be a nonabelian group. Let G have order 4 and let G have ord...
Non-Abelian Group of Order 4 and Order 3
Assuming that G is a non-abelian group with order 4 and order 3, we can deduce the following:
- The order of G is not 12, since the order of a group is the product of the orders of its cyclic subgroups, and G cannot have a cyclic subgroup of order 12.
- The order of G is not 6, since the only groups of order 6 are cyclic or isomorphic to S3, which are both abelian.
- Therefore, the only possibility is that the order of G is 4, and it has a subgroup of order 3.
Order of the Element in G
Based on the above observations, we cannot determine the order of the element in G. It could be finite or infinite, and there is no restriction on its form.
Option D is the Correct Answer
Hence, the correct answer is option D, which states that the order of the element in G need not be finite. This is because there are infinite groups that satisfy the given conditions, and they can have elements of any order, including infinite orders.
Conclusion
In conclusion, the order of the element in a non-abelian group G with order 4 and order 3 cannot be determined solely based on its properties. The answer could be finite or infinite, and there is no restriction on its form. Option D is the correct answer.
Assuming that G is a non-abelian group with order 4 and order 3, we can deduce the following:
- The order of G is not 12, since the order of a group is the product of the orders of its cyclic subgroups, and G cannot have a cyclic subgroup of order 12.
- The order of G is not 6, since the only groups of order 6 are cyclic or isomorphic to S3, which are both abelian.
- Therefore, the only possibility is that the order of G is 4, and it has a subgroup of order 3.
Order of the Element in G
Based on the above observations, we cannot determine the order of the element in G. It could be finite or infinite, and there is no restriction on its form.
Option D is the Correct Answer
Hence, the correct answer is option D, which states that the order of the element in G need not be finite. This is because there are infinite groups that satisfy the given conditions, and they can have elements of any order, including infinite orders.
Conclusion
In conclusion, the order of the element in a non-abelian group G with order 4 and order 3 cannot be determined solely based on its properties. The answer could be finite or infinite, and there is no restriction on its form. Option D is the correct answer.
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Let G be a nonabelian group. Let G have order 4 and let G have order 3. Then the order of the element in Ga)is 6b)is 12c)is of the form 12k for k 2d)need not be finiteCorrect answer is option 'D'. Can you explain this answer?
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Let G be a nonabelian group. Let G have order 4 and let G have order 3. Then the order of the element in Ga)is 6b)is 12c)is of the form 12k for k 2d)need not be finiteCorrect answer is option 'D'. 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 Let G be a nonabelian group. Let G have order 4 and let G have order 3. Then the order of the element in Ga)is 6b)is 12c)is of the form 12k for k 2d)need not be finiteCorrect answer is option 'D'. 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 Let G be a nonabelian group. Let G have order 4 and let G have order 3. Then the order of the element in Ga)is 6b)is 12c)is of the form 12k for k 2d)need not be finiteCorrect answer is option 'D'. Can you explain this answer?.
Let G be a nonabelian group. Let G have order 4 and let G have order 3. Then the order of the element in Ga)is 6b)is 12c)is of the form 12k for k 2d)need not be finiteCorrect answer is option 'D'. 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 Let G be a nonabelian group. Let G have order 4 and let G have order 3. Then the order of the element in Ga)is 6b)is 12c)is of the form 12k for k 2d)need not be finiteCorrect answer is option 'D'. 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 Let G be a nonabelian group. Let G have order 4 and let G have order 3. Then the order of the element in Ga)is 6b)is 12c)is of the form 12k for k 2d)need not be finiteCorrect answer is option 'D'. Can you explain this answer?.
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Let G be a nonabelian group. Let G have order 4 and let G have order 3. Then the order of the element in Ga)is 6b)is 12c)is of the form 12k for k 2d)need not be finiteCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for Let G be a nonabelian group. Let G have order 4 and let G have order 3. Then the order of the element in Ga)is 6b)is 12c)is of the form 12k for k 2d)need not be finiteCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of Let G be a nonabelian group. Let G have order 4 and let G have order 3. Then the order of the element in Ga)is 6b)is 12c)is of the form 12k for k 2d)need not be finiteCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
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