Disjoint cycles refer to cycles in a permutation that have no common elements. In other words, they do not share any numbers in their cycle notation.
The numbers inside the parentheses represent the elements that are being permuted. The numbers are written in a cycle notation, where the first number is mapped to the second number, the second to the third, and so on. The last number is mapped back to the first number.
To write (13257)(23)(47512) as a product of disjoint cycles, you need to break it down into smaller cycles. First, write down the numbers in their original order: 1234567. Then, identify the elements that are being permuted together. In this case, 1 is mapped to 3, 3 is mapped to 2, 2 is mapped to 5, 5 is mapped to 7, and 7 is mapped to 1. This forms the cycle (13752). Similarly, the cycle (23) maps 2 to 3 and 3 back to 2. The cycle (47512) maps 4 to 7, 7 to 5, 5 to 1, 1 to 4, and 2 to 2 (since it is not being permuted). Therefore, the product of disjoint cycles is (13752)(23)(47512).
Writing permutations as a product of disjoint cycles helps to simplify the notation and make it easier to understand. It also allows us to identify the order of the permutation, which is the smallest number of times the permutation needs to be applied to return to the original order.
There is no specific order in which the cycles should be written. However, it is common practice to write the cycles in descending order of length, starting with the longest cycle. In this case, it would be (13752)(47512)(23). But, as long as the cycles are disjoint, the order does not affect the final product.