Mathematics Exam > Mathematics Questions > Which of the following is true?a)The set of a...
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Which of the following is true?
- a)The set of all 2 x 2 real matrices forms a group under matrix multiplication
- b)A finite abelian group of order 6 has exactly two non-trivial subgroups
- c)Every finite group is always cyclic '
- d)The set of all 2 x 2 real non-singular matrices forms an abelian group under matrix multiplication
Correct answer is option 'B'. Can you explain this answer?
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Which of the following is true?a)The set of all 2 x 2 real matrices fo...
If we take matrix A = 
then A-1 does not exist.
Hence, option (a) discarded.
If we take a finite group S3 which is not cyclic the option (e) discarded.
Also , the set o f all 2 x 2 real non singular matrices may or may not form an abelian group under matrix multiplication.
Hence, option (d) discarded.
Hence, a finite abelian group of order 6 has exactly two non-trivial subgroups, i.e. a finite abelian group of order 6 is isomorphic to Z6 and Z6 is cyclic and it has exactly two non-trivial subgroups.
then A-1 does not exist.Hence, option (a) discarded.
If we take a finite group S3 which is not cyclic the option (e) discarded.
Also , the set o f all 2 x 2 real non singular matrices may or may not form an abelian group under matrix multiplication.
Hence, option (d) discarded.
Hence, a finite abelian group of order 6 has exactly two non-trivial subgroups, i.e. a finite abelian group of order 6 is isomorphic to Z6 and Z6 is cyclic and it has exactly two non-trivial subgroups.
Most Upvoted Answer
Which of the following is true?a)The set of all 2 x 2 real matrices fo...
Explanation:
To determine which of the given options is true, let's analyze each option one by one.
Option a: The set of all 2 x 2 real matrices does not form a group under matrix multiplication. Matrix multiplication is not commutative, meaning that AB does not always equal BA for matrices A and B. Therefore, the set of all 2 x 2 real matrices does not satisfy the group property of closure and is not a group.
Option b: A finite abelian group of order 6 has exactly two non-trivial subgroups. To prove this, we can use Lagrange's theorem, which states that the order of any subgroup of a finite group divides the order of the group.
Since the order of the group is 6, the possible orders of the subgroups are 1, 2, 3, and 6. The trivial subgroup, which contains only the identity element, has order 1.
If there is a subgroup of order 6, it would be the entire group itself. However, this is not possible as the order of a subgroup must divide the order of the group.
If there are subgroups of order 2, they must be cyclic and contain the identity element and an element of order 2. This subgroup can be generated by any element of order 2 in the group. There are three elements of order 2 in the group, so there are three subgroups of order 2.
If there are subgroups of order 3, they must also be cyclic and contain the identity element and an element of order 3. This subgroup can be generated by any element of order 3 in the group. There are two elements of order 3 in the group, so there are two subgroups of order 3.
Therefore, we have a total of 1 + 3 + 2 = 6 subgroups, two of which are non-trivial (subgroups of order 2 and order 3).
Option c: Every finite group is not always cyclic. There exist finite groups that are not cyclic, such as the symmetric group on four elements (S4). S4 is a non-cyclic group of order 24.
Option d: The set of all 2 x 2 real non-singular matrices does not form an abelian group under matrix multiplication. Matrix multiplication is not commutative, so the group does not satisfy the commutative property required for an abelian group.
Therefore, the correct answer is option b, as a finite abelian group of order 6 has exactly two non-trivial subgroups.
To determine which of the given options is true, let's analyze each option one by one.
Option a: The set of all 2 x 2 real matrices does not form a group under matrix multiplication. Matrix multiplication is not commutative, meaning that AB does not always equal BA for matrices A and B. Therefore, the set of all 2 x 2 real matrices does not satisfy the group property of closure and is not a group.
Option b: A finite abelian group of order 6 has exactly two non-trivial subgroups. To prove this, we can use Lagrange's theorem, which states that the order of any subgroup of a finite group divides the order of the group.
Since the order of the group is 6, the possible orders of the subgroups are 1, 2, 3, and 6. The trivial subgroup, which contains only the identity element, has order 1.
If there is a subgroup of order 6, it would be the entire group itself. However, this is not possible as the order of a subgroup must divide the order of the group.
If there are subgroups of order 2, they must be cyclic and contain the identity element and an element of order 2. This subgroup can be generated by any element of order 2 in the group. There are three elements of order 2 in the group, so there are three subgroups of order 2.
If there are subgroups of order 3, they must also be cyclic and contain the identity element and an element of order 3. This subgroup can be generated by any element of order 3 in the group. There are two elements of order 3 in the group, so there are two subgroups of order 3.
Therefore, we have a total of 1 + 3 + 2 = 6 subgroups, two of which are non-trivial (subgroups of order 2 and order 3).
Option c: Every finite group is not always cyclic. There exist finite groups that are not cyclic, such as the symmetric group on four elements (S4). S4 is a non-cyclic group of order 24.
Option d: The set of all 2 x 2 real non-singular matrices does not form an abelian group under matrix multiplication. Matrix multiplication is not commutative, so the group does not satisfy the commutative property required for an abelian group.
Therefore, the correct answer is option b, as a finite abelian group of order 6 has exactly two non-trivial subgroups.
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Which of the following is true?a)The set of all 2 x 2 real matrices forms a group under matrix multiplicationb)A finite abelian group of order 6 has exactly two non-trivial subgroupsc)Every finite group is always cyclic 'd)The set of all 2 x 2 real non-singular matrices forms an abelian group under matrix multiplicationCorrect answer is option 'B'. Can you explain this answer?
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Which of the following is true?a)The set of all 2 x 2 real matrices forms a group under matrix multiplicationb)A finite abelian group of order 6 has exactly two non-trivial subgroupsc)Every finite group is always cyclic 'd)The set of all 2 x 2 real non-singular matrices forms an abelian group under matrix multiplicationCorrect 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 true?a)The set of all 2 x 2 real matrices forms a group under matrix multiplicationb)A finite abelian group of order 6 has exactly two non-trivial subgroupsc)Every finite group is always cyclic 'd)The set of all 2 x 2 real non-singular matrices forms an abelian group under matrix multiplicationCorrect 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 true?a)The set of all 2 x 2 real matrices forms a group under matrix multiplicationb)A finite abelian group of order 6 has exactly two non-trivial subgroupsc)Every finite group is always cyclic 'd)The set of all 2 x 2 real non-singular matrices forms an abelian group under matrix multiplicationCorrect answer is option 'B'. Can you explain this answer?.
Which of the following is true?a)The set of all 2 x 2 real matrices forms a group under matrix multiplicationb)A finite abelian group of order 6 has exactly two non-trivial subgroupsc)Every finite group is always cyclic 'd)The set of all 2 x 2 real non-singular matrices forms an abelian group under matrix multiplicationCorrect 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 true?a)The set of all 2 x 2 real matrices forms a group under matrix multiplicationb)A finite abelian group of order 6 has exactly two non-trivial subgroupsc)Every finite group is always cyclic 'd)The set of all 2 x 2 real non-singular matrices forms an abelian group under matrix multiplicationCorrect 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 true?a)The set of all 2 x 2 real matrices forms a group under matrix multiplicationb)A finite abelian group of order 6 has exactly two non-trivial subgroupsc)Every finite group is always cyclic 'd)The set of all 2 x 2 real non-singular matrices forms an abelian group under matrix multiplicationCorrect answer is option 'B'. Can you explain this answer?.
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