Question

what is the permutation of s3

Answer 1

The symmetric group \( S_3 \) represents the group of all permutations of three elements. The group \( S_3 \) contains 6 elements (since there are \( 3! = 6 \) permutations of 3 items).

The permutations of \( S_3 \) can be explicitly listed as follows, assuming we are permuting the elements \( \{1, 2, 3\} \):

1. \( e \) (the identity permutation): \( (1, 2, 3) \)
2. \( (1, 2) \): swaps 1 and 2, resulting in \( (2, 1, 3) \)
3. \( (1, 3) \): swaps 1 and 3, resulting in \( (3, 2, 1) \)
4. \( (2, 3) \): swaps 2 and 3, resulting in \( (1, 3, 2) \)
5. \( (1, 2, 3) \): cyclically permutes to \( (2, 3, 1) \)
6. \( (1, 3, 2) \): cyclically permutes to \( (3, 1, 2) \)

In cycle notation, we can represent the permutations of \( S_3 \) as follows:

1. \( e = () \) (the identity)
2. \( (1\ 2) \)
3. \( (1\ 3) \)
4. \( (2\ 3) \)
5. \( (1\ 2\ 3) \) (a cycle permutation)
6. \( (1\ 3\ 2) \) (the reverse cycle)

These permutations cover all possible arrangements of the three elements.