The symmetric group S4 contains exactly 24 elements. This number represents all possible unique ways to arrange a set of four distinct items.
Understanding the Symmetric Group S4
The symmetric group Sn, also known as the permutation group on n elements, is the group whose elements are all possible permutations of n distinct objects, and whose group operation is the composition of these permutations. Each element in Sn is a bijective mapping (a bijection) from the set {1, 2, ..., n} to itself.
For S4, we are considering permutations of a set with four distinct elements, for instance, {1, 2, 3, 4}.
Calculating the Number of Elements
The total number of elements in any symmetric group Sn is given by the factorial of n, denoted as n!. This is because for the first position, there are n choices, for the second position there are n-1 choices remaining, for the third there are n-2, and so on, until only 1 choice remains for the last position.
For S4, the number of elements is calculated as follows:
- 4! = 4 × 3 × 2 × 1 = 24
These 24 elements are unique arrangements or bijections of four objects. They represent every possible way to rearrange four items, from leaving them in their original order to swapping them in various configurations.
Factorial Values for Small Symmetric Groups
To provide context, here's how the number of elements (the order of the group) grows for small symmetric groups:
| Symmetric Group | Number of Elements (n!) |
|---|---|
| S1 | 1! = 1 |
| S2 | 2! = 2 × 1 = 2 |
| S3 | 3! = 3 × 2 × 1 = 6 |
| S4 | 4! = 4 × 3 × 2 × 1 = 24 |
| S5 | 5! = 5 × 4 × 3 × 2 × 1 = 120 |
Examples of Permutations in S4
Each of the 24 elements in S4 is a specific rearrangement. For instance, if we label the elements as 1, 2, 3, 4:
- (1 2 3 4): The identity permutation, where nothing changes.
- (1 2): Swaps elements 1 and 2, leaving 3 and 4 in place.
- (1 2 3): Moves 1 to 2, 2 to 3, and 3 to 1, leaving 4 in place.
- (1 2)(3 4): Swaps 1 and 2, AND also swaps 3 and 4 simultaneously.
These are just a few examples; in total, there are 24 distinct permutations that form the group S4.
Significance of S4
The symmetric group S4 is a fundamental object in abstract algebra. It is a non-abelian group, meaning that the order in which permutations are composed matters (e.g., (1 2) followed by (2 3) is not the same as (2 3) followed by (1 2)). Understanding its structure and elements is crucial for studying group theory, its applications in fields like crystallography, quantum mechanics, and even in puzzles like the Rubik's Cube, where permutations play a central role.
