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Let S9 be the group of all permutations of the set {1, 2, 3, 4, 5, 6, 7, 8, 9} Then the total number of elements in S9 that commute with tau = (123)(4567) in S9 equals
(a) 21
(b) 22
(c) 24
(d) 25?
(a) 21
(b) 22
(c) 24
(d) 25?
Most Upvoted Answer
Let S9 be the group of all permutations of the set {1, 2, 3, 4, 5, 6, ...
Understanding the Problem
To find the elements in S9 that commute with the permutation tau = (123)(4567), we first analyze the structure of tau.
Structure of tau
- Cycles in tau:
- (123) is a 3-cycle affecting the elements {1, 2, 3}.
- (4567) is a 4-cycle affecting the elements {4, 5, 6, 7}.
- Fixed Points:
- Elements {8, 9} remain unchanged.
Commuting Condition
A permutation σ in S9 commutes with tau if σ * tau = tau * σ. This condition requires σ to preserve the cycle structure of tau.
Elements in S9 that Commute with tau
1. Permutations of {1, 2, 3}:
- The permutations of the 3-cycle (123) that preserve the cycle structure are:
- The identity (1)
- The 3 rotations: (123), (132)
- Total = 3 permutations.
2. Permutations of {4, 5, 6, 7}:
- The permutations of the 4-cycle (4567) that preserve the cycle structure are:
- The identity (1)
- The 4 rotations: (4567), (4675), (4756), (4756)
- Total = 4 permutations.
3. Fixed Points {8, 9}:
- Elements {8, 9} can be permuted freely with 2 possible arrangements:
- The identity (1)
- The swap (89)
- Total = 2 permutations.
Calculating Total Commuting Elements
The total number of permutations commuting with tau can be calculated by multiplying the number of options for each cycle:
- Total = 3 (from {1, 2, 3}) * 4 (from {4, 5, 6, 7}) * 2 (from {8, 9}) = 24.
Conclusion
Thus, the number of elements in S9 that commute with tau = (123)(4567) is 24.
The answer is (c) 24.
To find the elements in S9 that commute with the permutation tau = (123)(4567), we first analyze the structure of tau.
Structure of tau
- Cycles in tau:
- (123) is a 3-cycle affecting the elements {1, 2, 3}.
- (4567) is a 4-cycle affecting the elements {4, 5, 6, 7}.
- Fixed Points:
- Elements {8, 9} remain unchanged.
Commuting Condition
A permutation σ in S9 commutes with tau if σ * tau = tau * σ. This condition requires σ to preserve the cycle structure of tau.
Elements in S9 that Commute with tau
1. Permutations of {1, 2, 3}:
- The permutations of the 3-cycle (123) that preserve the cycle structure are:
- The identity (1)
- The 3 rotations: (123), (132)
- Total = 3 permutations.
2. Permutations of {4, 5, 6, 7}:
- The permutations of the 4-cycle (4567) that preserve the cycle structure are:
- The identity (1)
- The 4 rotations: (4567), (4675), (4756), (4756)
- Total = 4 permutations.
3. Fixed Points {8, 9}:
- Elements {8, 9} can be permuted freely with 2 possible arrangements:
- The identity (1)
- The swap (89)
- Total = 2 permutations.
Calculating Total Commuting Elements
The total number of permutations commuting with tau can be calculated by multiplying the number of options for each cycle:
- Total = 3 (from {1, 2, 3}) * 4 (from {4, 5, 6, 7}) * 2 (from {8, 9}) = 24.
Conclusion
Thus, the number of elements in S9 that commute with tau = (123)(4567) is 24.
The answer is (c) 24.
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Let S9 be the group of all permutations of the set {1, 2, 3, 4, 5, 6, 7, 8, 9} Then the total number of elements in S9 that commute with tau = (123)(4567) in S9 equals(a) 21(b) 22 (c) 24(d) 25?
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Let S9 be the group of all permutations of the set {1, 2, 3, 4, 5, 6, 7, 8, 9} Then the total number of elements in S9 that commute with tau = (123)(4567) in S9 equals(a) 21(b) 22 (c) 24(d) 25? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let S9 be the group of all permutations of the set {1, 2, 3, 4, 5, 6, 7, 8, 9} Then the total number of elements in S9 that commute with tau = (123)(4567) in S9 equals(a) 21(b) 22 (c) 24(d) 25? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let S9 be the group of all permutations of the set {1, 2, 3, 4, 5, 6, 7, 8, 9} Then the total number of elements in S9 that commute with tau = (123)(4567) in S9 equals(a) 21(b) 22 (c) 24(d) 25?.
Let S9 be the group of all permutations of the set {1, 2, 3, 4, 5, 6, 7, 8, 9} Then the total number of elements in S9 that commute with tau = (123)(4567) in S9 equals(a) 21(b) 22 (c) 24(d) 25? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let S9 be the group of all permutations of the set {1, 2, 3, 4, 5, 6, 7, 8, 9} Then the total number of elements in S9 that commute with tau = (123)(4567) in S9 equals(a) 21(b) 22 (c) 24(d) 25? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let S9 be the group of all permutations of the set {1, 2, 3, 4, 5, 6, 7, 8, 9} Then the total number of elements in S9 that commute with tau = (123)(4567) in S9 equals(a) 21(b) 22 (c) 24(d) 25?.
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