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The question asks to list all the permutations of \( S_4 \), where \( S_4 \) refers to the symmetric group on 4 elements. Symmetric groups consist of all the possible permutations (rearrangements) of a given set of elements. For \( S_4 \), the elements are typically \( \{1, 2, 3, 4\} \).

There are \( 4! = 24 \) permutations in \( S_4 \). Let's list them:

1. \( (1\ 2\ 3\ 4) \)
2. \( (1\ 2\ 4\ 3) \)
3. \( (1\ 3\ 2\ 4) \)
4. \( (1\ 3\ 4\ 2) \)
5. \( (1\ 4\ 2\ 3) \)
6. \( (1\ 4\ 3\ 2) \)
7. \( (2\ 1\ 3\ 4) \)
8. \( (2\ 1\ 4\ 3) \)
9. \( (2\ 3\ 1\ 4) \)
10. \( (2\ 3\ 4\ 1) \)
11. \( (2\ 4\ 1\ 3) \)
12. \( (2\ 4\ 3\ 1) \)
13. \( (3\ 1\ 2\ 4) \)
14. \( (3\ 1\ 4\ 2) \)
15. \( (3\ 2\ 1\ 4) \)
16. \( (3\ 2\ 4\ 1) \)
17. \( (3\ 4\ 1\ 2) \)
18. \( (3\ 4\ 2\ 1) \)
19. \( (4\ 1\ 2\ 3) \)
20. \( (4\ 1\ 3\ 2) \)
21. \( (4\ 2\ 1\ 3) \)
22. \( (4\ 2\ 3\ 1) \)
23. \( (4\ 3\ 1\ 2) \)
24. \( (4\ 3\ 2\ 1) \)

These are all the permutations in \( S_4 \). Each corresponds to a unique way of arranging the four elements \( \{1, 2, 3, 4\} \).

Let me know if you'd like further explanations on these or other related questions. Would you like to see details on how we calculate the number of permutations or something else?

### Relative Questions:
1. How do we calculate the total number of elements in \( S_n \)?
2. What is the general structure of a symmetric group?
3. Can you show examples of cycles and transpositions in \( S_4 \)?
4. How do you determine the order of an element in \( S_4 \)?
5. How are permutation groups applied in real-world contexts?

### Tip:
Remember that the symmetric group \( S_n \) consists of all possible bijections (permutations) of a set with \( n \) elements, and its order is \( n! \) (factorial of \( n \)).

Let S_4 be the symmetric group. List all the permutations on S_4.

Explore the complete list of 24 permutations in the symmetric group S_4. This problem introduces concepts from group theory, covering how to list permutations and calculate the number of elements in S_n using factorials. Suitable for undergraduate students studying abstract algebra.

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