Elements of permutation group of 4
The permutation group of 4, or Symmetric Group S4, consists of all possible permutations of 4 objects. There are 4! = 24 possible permutations in this group.
The elements of this group can be represented by permutations of the numbers 1, 2, 3, and 4. For example, the identity permutation is (1)(2)(3)(4), which means that all 4 objects remain in their original positions.
Some other elements of S4 are:
- (1 2), which means that objects 1 and 2 swap places, while objects 3 and 4 remain in their original positions.
- (1 4), which means that objects 1 and 4 swap places, while objects 2 and 3 remain in their original positions.
- (1 2)(3 4), which means that objects 1 and 2 swap places, and objects 3 and 4 swap places.
There are 20 more permutations in this group, but these are some examples of the types of elements that are included.
Include all elements of the group
Sure, here are all the 24 elements of S4, represented as permutations of the numbers 1, 2, 3, and 4:
(1)(2)(3)(4) - identity permutation
(1 2)(3)(4)
(1 3)(2)(4)
(1 4)(2)(3)
(1)(2 3)(4)
(1)(2 4)(3)
(1)(3 4)(2)
(1 2 3)(4)
(1 2 4)(3)
(1 3 2)(4)
(1 3 4)(2)
(1 4 2)(3)
(1 4 3)(2)
(1 2)(3 4)
(1 3)(2 4)
(1 4)(2 3)
(1)(2 3 4)
(1)(2 4 3)
(1)(3 2 4)
(1)(3 4 2)
(1)(4 2 3)
(1)(4 3 2)
(1 2 3 4)
(1 4 3 2) - permutation with all elements reversed
Note that the group operation in S4 is composition of permutations, so each element is represented by a product of transpositions (permutations that swap two elements). For example, (1 2 3) can be written as (1 2)(2 3).
The elements of the permutation group of 4, denoted as S4, are all possible permutations of the set {1, 2, 3, 4}. To determine the elements, we can use a systematic approach.
1. Start with the identity permutation, where every element is mapped to itself: (1)(2)(3)(4).
2. Consider the permutations that swap two elements. There are six possible pairs of elements to swap: (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), and (3, 4). Each pair can either be swapped or not, resulting in 2^6 = 64 permutations. However, two of these permutations are already covered by the identity permutation, so we exclude them.
3. Continue with permutations that involve three elements. There are four possible triplets of elements: (1, 2, 3), (1, 2, 4), (1, 3, 4), and (2, 3, 4). Each triplet can be permuted in 3! = 6 different ways.
4. Finally, consider permutations that involve all four elements. There is only one permutation of (1, 2, 3, 4) since all elements must be mapped to each other.
In total, the permutation group S4 consists of 24 permutations:
(1)(2)(3)(4)
(1 2)(3)(4)
(1 3)(2)(4)
(1 4)(2)(3)
(2 3)(1)(4)
(2 4)(1)(3)
(3 4)(1)(2)
(1 2 3)(4)
(1 2 4)(3)
(1 3 4)(2)
(2 3 4)(1)
(1 3 2)(4)
(1 4 3)(2)
(2 4 3)(1)
(2 1)(3)(4)
(3 1)(2)(4)
(4 1)(2)(3)
(3 2)(1)(4)
(4 2)(1)(3)
(4 3)(1)(2)
(1 2)(4)(3)
(1 3)(4)(2)
(1 4)(3)(2)
(2 3)(4)(1)
(2 4)(3)(1)
(3 4)(2)(1)