Understanding the Group S4
S4, or the symmetric group on 4 elements, consists of all the permutations of a set with four elements. The order of S4 is 4! (which equals 24). To find the elements of order two, we need to identify the permutations that, when applied twice, return the original arrangement.
Elements of Order Two
In S4, the permutations of order two can be characterized as follows:
- Transpositions: These are simple swaps of two elements, leaving the other two unchanged. The general form of a transposition in S4 is (a b), where a and b are distinct elements from the set {1, 2, 3, 4}.
- Products of Two Disjoint Transpositions: This consists of swapping two pairs of elements simultaneously. An example is (a b)(c d), where (a b) and (c d) are both transpositions.
Counting the Elements
1. Transpositions:
- The number of ways to choose 2 elements from 4 can be calculated using combinations.
- There are C(4, 2) = 6 transpositions: (1 2), (1 3), (1 4), (2 3), (2 4), and (3 4).
2. Products of Two Disjoint Transpositions:
- To form a product of two disjoint transpositions, we can select 4 elements and pair them.
- There are C(4, 2)/2 = 3 such products: (1 2)(3 4), (1 3)(2 4), and (1 4)(2 3). The division by 2 accounts for the fact that the order of the pairs does not matter.
Total Elements of Order Two
- Adding the counts together gives us a total of:
- 6 (from transpositions) + 3 (from products of two transpositions) = 9.
Thus, the number of elements of order two in the group S4 is 9.