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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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#3. Thanks.
![### 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbb66fa7-7c22-4982-a22f-aaed542f65b3%2F0565aedf-8845-4e51-89f2-7ccda2d4c5e5%2Fbmotly_processed.png&w=3840&q=75)
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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