pi=(245)(1354)(125)
1) Write as a product of disjoint
cycles.
Ans: (14)(253) Is this correct?
2) Solve : pi^2, pi^5,pi^(-1)
3) What's the order of pi? and why?
To write the permutation pi as a product of disjoint cycles, we need to examine how each element gets mapped by the permutation.
The given permutation pi can be written as: pi = (245)(1354)(125)
1) Writing as a product of disjoint cycles:
To do this, we look at each number from 1 to the maximum number present in the given permutation. We follow the cycle of each number until we reach the original number again. Let's go step by step:
- Starting with the number 1:
- 1 is mapped to 2 (in the first cycle)
- 2 is mapped to 4 (in the first cycle)
- 4 is mapped back to 1 (in the first cycle)
Therefore, the cycle for 1 is (124)
- Moving to the next number 2:
- 2 is already part of the previous cycle, so we move to the next number.
- Skipping 3 as it is not in any of the cycles.
- Moving to the next number 4:
- 4 is already part of the previous cycle, so we move to the next number.
- Moving to the next number 5:
- 5 is mapped to 3 (in the second cycle)
- 3 is mapped to 1 (in the second cycle)
- 1 is already part of the previous cycle, so we move to the next number.
- Moving to the next number 3:
- 3 is already part of the previous cycle, so we move to the next number.
- Moving to the next number 1:
- 1 is already part of the previous cycle, so we move to the next number.
- Moving to the next number 2:
- 2 is already part of the previous cycle, so we move to the next number.
- Moving to the next number 5:
- 5 is already part of the previous cycle, so we move to the next number.
- Moving to the next number 4:
- 4 is already part of the previous cycle, so we move to the next number.
- Moving to the next number 5:
- 5 is already part of the previous cycle, so we move to the next number.
Hence, the product of disjoint cycles for the given permutation pi is: (124)(35)
2) Solving pi^2, pi^5, pi^(-1):
To solve these, we need to take each cycle in the permutation to the corresponding power. Let's calculate:
- pi^2:
(124)(35) ^2 = (124)(35)(124)(35)
(124)(35)(124)(35) = (1254)(34)
- pi^5:
(124)(35) ^5 = (124)(35)(124)(35)(124)(35)(124)(35)(124)(35)
(124)(35)(124)(35)(124)(35)(124)(35)(124)(35) = (14)(253)
- pi^(-1):
The inverse of a permutation is obtained by reversing each cycle. So, to find pi^(-1), we simply reverse the cycles in the product of disjoint cycles. In this case, reversing (124)(35) gives (142)(53).
Hence, pi^2 = (1254)(34), pi^5 = (14)(253), and pi^(-1) = (142)(53).
3) Finding the order of pi:
The order of a permutation is defined as the smallest positive integer that, when taken to that power, gives the identity permutation (pi^k = e, where e is the identity permutation).
To find the order of pi, we need to calculate pi, pi^2, pi^3, pi^4,... until we obtain the identity permutation.
- pi^2 = (1254)(34)
- pi^3 = (1254)(34)(245)(1354)(125) = (14)(253)(245)(1354)(125) = (14253)
- pi^4 = (14)(253)(245)(1354)(125)(245)(1354)(125) = (24513)
- pi^5 = (14)(253)
- pi^6 = (14)(253)(14253) = (1254)(34)(14253) = (124)(35)(14253) = (35)(124) = pi^(-1) (Inverse of pi)
So, we can see that pi^6 gives us the identity permutation (pi^6 = pi^(-1)). Therefore, the order of pi is 6.
Explanation for finding the order:
We found the order of pi by calculating the powers of pi and observing when we obtain the identity permutation. In this case, it took 6 powers of pi to reach the identity permutation.