How do we find even permutations of S3

Answer 1

The symmetric group S3 has only six permutations, denoted as follows:

1. (1) - the identity permutation
2. (12)
3. (13)
4. (23)
5. (123)
6. (132)

An even permutation is a permutation that can be expressed as the product of an even number of transpositions (i.e. swapping two elements). In S3, the only even permutations are the identity permutation and the permutations (12)(13) and (23)(13). This is because:

- (12) and (23) are odd permutations, as each is a product of a single transposition.
- (123) and (132) are odd permutations, as each is a product of three transpositions.
- (13) and (23) are odd permutations, as each is a product of two transpositions (i.e. (12)(13)(12) and (12)(23)(12), respectively).

Therefore, the even permutations of S3 are:

1. (1) - the identity permutation
2. (12)(13)
3. (23)(13)

Answer 2

Express each element of S3 as a product of permutations

Answer 3

The symmetric group S3 has six elements, and each element can be expressed as a product of permutations. Here are all the possible permutations that can be used to express each element of S3:

1. (1) - the identity permutation - this is already in its simplest form, and can't be expressed as a product of other permutations.
2. (12)
- (12) itself is already an irreducible cycle, and can't be expressed as a product of other permutations.
3. (13)
- (13) itself is already an irreducible cycle, and can't be expressed as a product of other permutations.
4. (23)
- (23) itself is already an irreducible cycle, and can't be expressed as a product of other permutations.
5. (123) - this can be expressed as the product of (12) and (13), which gives (123) = (12)(13).
6. (132) - this can be expressed as the product of (12) and (23), which gives (132) = (12)(23).

So, in summary:

- (1) = (1)
- (12) = (12)
- (13) = (13)
- (23) = (23)
- (123) = (12)(13)
- (132) = (12)(23)