How do we find even permutations of S3
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)
Express each element of S3 as a product of permutations
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)
To find the even permutations of S3 (the symmetric group on 3 elements), you can follow these steps:
Step 1: Write down all the permutations of S3.
S3 consists of the set {1, 2, 3}, so the permutations can be written as:
(1, 2, 3), (1, 3, 2)
(2, 1, 3), (2, 3, 1)
(3, 1, 2), (3, 2, 1)
Step 2: Assign a sign to each permutation.
To determine if a permutation is even or odd, you need to count the number of inversions. An inversion occurs when a number is followed by a smaller number in the permutation. If the number of inversions is even, the permutation is considered even; if it is odd, the permutation is considered odd.
Take each permutation and count the number of inversions:
- (1, 2, 3): No inversions.
- (1, 3, 2): 3 follows both 1 and 2, so there are two inversions.
- (2, 1, 3): 1 follows 2, so there is one inversion.
- (2, 3, 1): No inversions.
- (3, 1, 2): 1 follows 3, and 2 follows 1 and 3, so there are three inversions.
- (3, 2, 1): 2 follows both 3 and 1, so there are two inversions.
Step 3: Determine the sign of each permutation.
Based on the number of inversions, determine if each permutation is even or odd.
In this case, the even permutations have an even number of inversions, and the odd permutations have an odd number of inversions.
Using the above counts, the signs of the permutations are as follows:
(1, 2, 3): Even
(1, 3, 2): Odd
(2, 1, 3): Odd
(2, 3, 1): Even
(3, 1, 2): Odd
(3, 2, 1): Even
So the even permutations of S3 are: (1, 2, 3), (2, 3, 1), and (3, 1, 2).
To find the even permutations of S3, we need to understand what S3 represents.
S3 refers to the symmetric group of order 3, which consists of all the possible permutations of three elements. In this case, the three elements are usually denoted as {1, 2, 3}.
To find even permutations of S3, we can follow these steps:
Step 1: List down all the possible permutations of S3.
- For S3, we can list all the permutations by simply rearranging the elements {1, 2, 3}.
- The possible permutations are:
{(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)}
Step 2: Identify which permutations are even.
- To determine if a permutation is even or odd, we use the concept of a "sign" or "parity" of a permutation.
- The sign of a permutation is determined by counting the number of element exchanges needed to get back to the original ordering.
- An even permutation has an even number of element exchanges, while an odd permutation has an odd number of element exchanges.
- In the case of S3, we can check the parity of each permutation using various methods, such as:
- By visual inspection: You can manually count the exchanges needed to return to the original ordering.
- By using the permutation's cycle notation: Write each permutation in cycle notation and count the number of even-length cycles.
- By using the permutation's matrix representation: Convert each permutation into a matrix and compute the determinant. If the determinant is -1, it is odd; if it is 1, it is even.
By applying these methods, you can determine that the even permutations of S3 are:
{(1, 2, 3), (1, 3, 2), (2, 1, 3), (3, 1, 2)}.
Note: In general, for symmetric groups of larger orders, the process of finding even permutations may become more complex.