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3. List all of the permutations of S4. Find each of the following sets: a) {0 € S4:0 (1) = 3} b) {0 € S4:0 (2) = 2} c) {0 € S4:0(1) = 3 and o(2) = 2} Are any of these sets subgroups of S4? Justify your conclusions! (5 from 5.4)

3. List all of the permutations of S4. Find each of the following sets: a) {0 € S4:0 (1) = 3} b) {0 € S4:0 (2) = 2} c) {0 € S4:0(1) = 3 and o(2) = 2} Are any of these sets subgroups of S4? Justify your conclusions! (5 from 5.4)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Permutations in the Symmetric Group \( S_4 \) This exercise involves finding specific subsets of permutations in the symmetric group \( S_4 \), which consists of all permutations of the set \(\{1, 2, 3, 4\}\). **Problem Statement:** List all of the permutations of \( S_4 \). Find each of the following sets: a) \(\{\sigma \in S_4 : \sigma(1) = 3\}\) b) \(\{\sigma \in S_4 : \sigma(2) = 2\}\) c) \(\{\sigma \in S_4 : \sigma(1) = 3 \text{ and } \sigma(2) = 2\}\) **Further Inquiry:** Are any of these sets subgroups of \( S_4 \)? Justify your conclusions! *(This question is referenced from exercise 5 in section 5.4 of your textbook.)* ### Detailed Explanation **a) Set \(\{\sigma \in S_4 : \sigma(1) = 3\}\):** This set includes all permutations where number 1 is mapped to 3. Examples include: - (3,2,1,4) - (3,1,4,2) - (3,4,2,1) - ...and so on. **b) Set \(\{\sigma \in S_4 : \sigma(2) = 2\}\):** This set includes all permutations where number 2 stays as 2. Some examples would be: - (1,2,3,4) - (3,2,4,1) - (4,2,1,3) - ...and so on. **c) Set \(\{\sigma \in S_4 : \sigma(1) = 3 \text{ and } \sigma(2) = 2\}\):** This set includes all permutations where number 1 is mapped to 3 and number 2 remains 2. Examples include: - (3,2,1,4) - (3,2,4,1) - ...and so on. **Subgroup Analysis:** To determine if any of these sets are subgroups of \( S_4 \), we need to verify the subgroup criteria: 1. The set must contain the
Transcribed Image Text:### Permutations in the Symmetric Group \( S_4 \) This exercise involves finding specific subsets of permutations in the symmetric group \( S_4 \), which consists of all permutations of the set \(\{1, 2, 3, 4\}\). **Problem Statement:** List all of the permutations of \( S_4 \). Find each of the following sets: a) \(\{\sigma \in S_4 : \sigma(1) = 3\}\) b) \(\{\sigma \in S_4 : \sigma(2) = 2\}\) c) \(\{\sigma \in S_4 : \sigma(1) = 3 \text{ and } \sigma(2) = 2\}\) **Further Inquiry:** Are any of these sets subgroups of \( S_4 \)? Justify your conclusions! *(This question is referenced from exercise 5 in section 5.4 of your textbook.)* ### Detailed Explanation **a) Set \(\{\sigma \in S_4 : \sigma(1) = 3\}\):** This set includes all permutations where number 1 is mapped to 3. Examples include: - (3,2,1,4) - (3,1,4,2) - (3,4,2,1) - ...and so on. **b) Set \(\{\sigma \in S_4 : \sigma(2) = 2\}\):** This set includes all permutations where number 2 stays as 2. Some examples would be: - (1,2,3,4) - (3,2,4,1) - (4,2,1,3) - ...and so on. **c) Set \(\{\sigma \in S_4 : \sigma(1) = 3 \text{ and } \sigma(2) = 2\}\):** This set includes all permutations where number 1 is mapped to 3 and number 2 remains 2. Examples include: - (3,2,1,4) - (3,2,4,1) - ...and so on. **Subgroup Analysis:** To determine if any of these sets are subgroups of \( S_4 \), we need to verify the subgroup criteria: 1. The set must contain the
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