Mathematics Exam > Mathematics Questions > Which of the following is correct?a)Let G = S...
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Which of the following is correct?
- a)Let G = Sn be the permutation group on n symbols then for all f.g∈G, (f.g)2 = f2-g2
- b)Let G = S3 and let A = {f ∈G:f2 = Identity} then A is a sub-group of G
- c)Let G = S4 let B = { σ ∈ S4, :σ4 = Identity but of identity} then B contains exactly 7 elements
- d)A. (∀n>3) has a self inverse element
Correct answer is option 'D'. Can you explain this answer?
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Which of the following is correct?a)Let G = Snbe the permutation group...
If n > 3, then σ = (1, 2)(3, 4) ∈ An also 0(σ) = 2
⇒ An ∀n > 3 has a self inverse element.
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Which of the following is correct?a)Let G = Snbe the permutation group...
Understanding the Options
To determine the correctness of the statements about permutation groups, let’s analyze each option:
Option A: Incorrect
- The statement claims that for all f, g in G, (f.g)² = f² - g².
- This is incorrect as the operation of permutations does not follow this algebraic property. The composition of permutations does not distribute over addition or subtraction.
Option B: Incorrect
- Here, A is defined as the set of elements in S3 such that f² = Identity.
- The only elements that satisfy this condition are the identity and the transpositions (which are self-inverse). However, the set does not contain all inverses necessary to be a subgroup, since it lacks closure under composition.
Option C: Incorrect
- In S4, the set B is defined as the permutations where σ⁴ = Identity but not the identity itself.
- Analyzing the permutations of order 4, we find that there are 9 such permutations (the 4-cycles), and thus the claim of exactly 7 elements is false.
Option D: Correct
- This option states that for all n > 3, there exists a self-inverse element.
- In the permutation group Sn, transpositions (which swap two elements and leave the rest unchanged) are self-inverse elements.
- For any n greater than 3, you can always form transpositions, such as (1 2), which confirms that self-inverse elements exist.
Conclusion
The correct answer is indeed option D. For n > 3, the existence of self-inverse elements (transpositions) in the permutation group is guaranteed, confirming the statement's validity.
To determine the correctness of the statements about permutation groups, let’s analyze each option:
Option A: Incorrect
- The statement claims that for all f, g in G, (f.g)² = f² - g².
- This is incorrect as the operation of permutations does not follow this algebraic property. The composition of permutations does not distribute over addition or subtraction.
Option B: Incorrect
- Here, A is defined as the set of elements in S3 such that f² = Identity.
- The only elements that satisfy this condition are the identity and the transpositions (which are self-inverse). However, the set does not contain all inverses necessary to be a subgroup, since it lacks closure under composition.
Option C: Incorrect
- In S4, the set B is defined as the permutations where σ⁴ = Identity but not the identity itself.
- Analyzing the permutations of order 4, we find that there are 9 such permutations (the 4-cycles), and thus the claim of exactly 7 elements is false.
Option D: Correct
- This option states that for all n > 3, there exists a self-inverse element.
- In the permutation group Sn, transpositions (which swap two elements and leave the rest unchanged) are self-inverse elements.
- For any n greater than 3, you can always form transpositions, such as (1 2), which confirms that self-inverse elements exist.
Conclusion
The correct answer is indeed option D. For n > 3, the existence of self-inverse elements (transpositions) in the permutation group is guaranteed, confirming the statement's validity.
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Which of the following is correct?a)Let G = Snbe the permutation group on n symbols then for all f.g∈G, (f.g)2= f2-g2b)Let G = S3and let A = {f∈G:f2= Identity} then A is a sub-group of Gc)Let G = S4let B = {σ ∈S4, :σ4= Identity but of identity} then B contains exactly 7 elementsd)A. (∀n>3) has a self inverse elementCorrect answer is option 'D'. Can you explain this answer?
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Which of the following is correct?a)Let G = Snbe the permutation group on n symbols then for all f.g∈G, (f.g)2= f2-g2b)Let G = S3and let A = {f∈G:f2= Identity} then A is a sub-group of Gc)Let G = S4let B = {σ ∈S4, :σ4= Identity but of identity} then B contains exactly 7 elementsd)A. (∀n>3) has a self inverse elementCorrect answer is option 'D'. 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 correct?a)Let G = Snbe the permutation group on n symbols then for all f.g∈G, (f.g)2= f2-g2b)Let G = S3and let A = {f∈G:f2= Identity} then A is a sub-group of Gc)Let G = S4let B = {σ ∈S4, :σ4= Identity but of identity} then B contains exactly 7 elementsd)A. (∀n>3) has a self inverse elementCorrect answer is option 'D'. 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 correct?a)Let G = Snbe the permutation group on n symbols then for all f.g∈G, (f.g)2= f2-g2b)Let G = S3and let A = {f∈G:f2= Identity} then A is a sub-group of Gc)Let G = S4let B = {σ ∈S4, :σ4= Identity but of identity} then B contains exactly 7 elementsd)A. (∀n>3) has a self inverse elementCorrect answer is option 'D'. Can you explain this answer?.
Which of the following is correct?a)Let G = Snbe the permutation group on n symbols then for all f.g∈G, (f.g)2= f2-g2b)Let G = S3and let A = {f∈G:f2= Identity} then A is a sub-group of Gc)Let G = S4let B = {σ ∈S4, :σ4= Identity but of identity} then B contains exactly 7 elementsd)A. (∀n>3) has a self inverse elementCorrect answer is option 'D'. 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 correct?a)Let G = Snbe the permutation group on n symbols then for all f.g∈G, (f.g)2= f2-g2b)Let G = S3and let A = {f∈G:f2= Identity} then A is a sub-group of Gc)Let G = S4let B = {σ ∈S4, :σ4= Identity but of identity} then B contains exactly 7 elementsd)A. (∀n>3) has a self inverse elementCorrect answer is option 'D'. 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 correct?a)Let G = Snbe the permutation group on n symbols then for all f.g∈G, (f.g)2= f2-g2b)Let G = S3and let A = {f∈G:f2= Identity} then A is a sub-group of Gc)Let G = S4let B = {σ ∈S4, :σ4= Identity but of identity} then B contains exactly 7 elementsd)A. (∀n>3) has a self inverse elementCorrect answer is option 'D'. Can you explain this answer?.
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