The notation (1 2)(3 4) in S4 refers to a permutation in the symmetric group S4. This permutation is composed of two cycles - one that swaps the elements 1 and 2, and another that swaps the elements 3 and 4. This means that when this permutation is applied to the elements of S4, 1 and 2 are interchanged, and 3 and 4 are interchanged.
The symmetric group S4 has 24 elements. This can be calculated using the formula n!, where n is the number of elements in the group. In this case, n=4, so 4!=24. This means that there are 24 different ways to arrange the elements 1, 2, 3, and 4 in a permutation.
The order of a permutation in a symmetric group is equal to the least common multiple of the lengths of its cycles. In the case of (1 2)(3 4), the lengths of the cycles are 2 and 2, so the order is 2. This means that when this permutation is applied to the elements of S4, it will take 2 iterations to return to the original arrangement.
To multiply permutations in S4, you first write them in cycle notation. Then, you combine the cycles from both permutations, taking into account the order of the elements. For example, to multiply (1 2)(3 4) and (2 3), you would combine the cycles (1 2)(2 3)(3 4). This would result in the permutation (1 2 3 4).
The identity element in S4 is the permutation that leaves all elements unchanged. In cycle notation, this would be written as (1)(2)(3)(4). In other words, it is a cycle of length 1 for each of the elements in the group. This permutation is often denoted as e or id.