Question 1205501: Express each of the given permutation of [ 1 2 3 4 5 6 7 8 ] as a product of disjoint cycles.
[1 2
3 6
3 4
4 5
5 6
1 7
7 8 8 2 ]
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website! To express a permutation as a product of disjoint cycles, we follow these steps:
1. **Start with an element:** Choose any element, say 1.
2. **Trace its cycle:** Follow the permutation to see where 1 goes. 1 goes to 2, 2 goes to 8, 8 goes to 7, 7 goes to 1. So, we have the cycle (1 2 8 7).
3. **Choose an unused element:** Choose an element not in the previous cycle, say 3.
4. **Trace its cycle:** 3 goes to 6, 6 goes to 5, 5 goes to 4, 4 goes to 3. So, we have the cycle (3 6 5 4).
Since all elements are now included in cycles, we have expressed the permutation as a product of disjoint cycles:
**(1 2 8 7)(3 6 5 4)**