Mathematics Exam > Mathematics Questions > Which of the following is/are true?a)For ever...
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Which of the following is/are true?
- a)For every n∈ℕ. There exist a unique subgroup of (ℂ'''·) of order n.
- b)Define S ={G is a subgroup of (ℂ'''·)|o(G) is finite}. Then the cardinality of S is countably infinite.
- c)There exist atleast one positive integerm such that (ℂ'''·) has m elemnts of order m.
- d)None of the above
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Which of the following is/are true?a)For every n∈. There exist a ...
Explanation of the Statements
Let’s analyze each statement to understand why option B is the correct answer.
a) Unique subgroup of order n
- This statement is false in general. According to group theory, a group may have multiple subgroups of the same order. For example, the symmetric group S3 has multiple subgroups of order 2. Therefore, it does not guarantee the existence of a unique subgroup of order n for every n in a group.
b) Countable cardinality of S
- This statement is true. The set S consists of all subgroups of finite order within a group. Since there are only countably many finite orders (1, 2, 3, ...), for each finite order, the number of subgroups can be finite or countably infinite. In many common groups like Z, S3, etc., the collection of finite groups is indeed countable. Thus, S is countably infinite.
c) Existence of positive integer m
- This statement is false. While there may be groups that have elements of every finite order, it's not guaranteed that there is a positive integer m such that the group contains m elements of order m. For instance, the trivial group has no elements of positive order.
d) None of the above
- This statement is also false because statement B is true.
Conclusion
Thus, the correct answer is option B, as it accurately describes the cardinality of the set of subgroups of finite order.
Let’s analyze each statement to understand why option B is the correct answer.
a) Unique subgroup of order n
- This statement is false in general. According to group theory, a group may have multiple subgroups of the same order. For example, the symmetric group S3 has multiple subgroups of order 2. Therefore, it does not guarantee the existence of a unique subgroup of order n for every n in a group.
b) Countable cardinality of S
- This statement is true. The set S consists of all subgroups of finite order within a group. Since there are only countably many finite orders (1, 2, 3, ...), for each finite order, the number of subgroups can be finite or countably infinite. In many common groups like Z, S3, etc., the collection of finite groups is indeed countable. Thus, S is countably infinite.
c) Existence of positive integer m
- This statement is false. While there may be groups that have elements of every finite order, it's not guaranteed that there is a positive integer m such that the group contains m elements of order m. For instance, the trivial group has no elements of positive order.
d) None of the above
- This statement is also false because statement B is true.
Conclusion
Thus, the correct answer is option B, as it accurately describes the cardinality of the set of subgroups of finite order.
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Community Answer
Which of the following is/are true?a)For every n∈. There exist a ...
Given (ℂ'''·) is a group then we know that
(i) For every positive integer n, there are exactly 
elements of order n.
elements of order n.(ii) For every positive integer n, there exist exactly one cyclic subgroup of (ℂ'''·).
(iii) G has countably infinite number of finite subgroups.
⇒ option (a) and (b) is ture
For option (c).
We know that G has exactly 
elements of order n and 
≠ n∀n ∈ ℕ
elements of order n and ≠ n∀n ∈ ℕ⇒ there does not exist any positive integer m such that G has m elements of order m.
⇒ option (c) is not true
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Which of the following is/are true?a)For every n∈. There exist a unique subgroup of (·) of order n.b)Define S ={G is a subgroup of (·)|o(G) is finite}. Then the cardinality of S is countably infinite.c)There exist atleast one positive integerm such that (·) has m elemnts of order m.d)None of the aboveCorrect answer is option 'B'. Can you explain this answer?
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Which of the following is/are true?a)For every n∈. There exist a unique subgroup of (·) of order n.b)Define S ={G is a subgroup of (·)|o(G) is finite}. Then the cardinality of S is countably infinite.c)There exist atleast one positive integerm such that (·) has m elemnts of order m.d)None of the aboveCorrect answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Which of the following is/are true?a)For every n∈. There exist a unique subgroup of (·) of order n.b)Define S ={G is a subgroup of (·)|o(G) is finite}. Then the cardinality of S is countably infinite.c)There exist atleast one positive integerm such that (·) has m elemnts of order m.d)None of the aboveCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following is/are true?a)For every n∈. There exist a unique subgroup of (·) of order n.b)Define S ={G is a subgroup of (·)|o(G) is finite}. Then the cardinality of S is countably infinite.c)There exist atleast one positive integerm such that (·) has m elemnts of order m.d)None of the aboveCorrect answer is option 'B'. Can you explain this answer?.
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