In S(5), let pi=(245)(1354)(125).
(a) Write pi as a product of disjoint cycles.
(b) Determine pi^2, pi^5, pi^(-1)
(c) What is the order of pi? Why?
(a) To write pi=(245)(1354)(125) as a product of disjoint cycles, we observe that the elements 1, 2, 3, 4, and 5 appear in the cycles (245), (1354), and (125). We can write pi as a product of disjoint cycles by tracing the elements through the cycles. Let's start with 1:
1 is mapped to 2 by the cycle (245), then 2 is mapped to 4 by the same cycle, and finally, 4 is mapped back to 1 by the cycle (125). Therefore, 1 goes to 4.
We can continue this process for the remaining numbers:
2 goes to 4, 4 goes to 5, and 5 goes to 2 through the cycles.
3 is not present in any of the cycles, so it remains unchanged.
Putting it all together, we have pi = (1452)(3)
(b) To determine pi^2, we need to apply the permutation pi to itself twice:
pi^2 = (1452)(3)(1452)(3)
For pi^5, we would apply pi to itself five times:
pi^5 = (1452)(3)(1452)(3)(1452)(3)(1452)(3)(1452)(3)
For pi^(-1), we need to find the inverse of each cycle and reverse their order:
pi^(-1) = (2541)(3)
(c) The order of pi is the smallest positive integer n such that pi^n = e (the identity element). To find the order, we can calculate pi^2, pi^3, pi^4, and so on until we reach the identity permutation.
Calculating:
pi^2 = (1452)(3)(1452)(3) = (154)(2)(35)
pi^3 = (1452)(3)(154)(2)(35) = (15)(42)(3)
pi^4 = (1452)(3)(15)(42)(3) = (1542)
pi^5 = (1452)(3)(1542) = (1)(2)(3)(4)(5) = e (identity element)
Therefore, the order of pi is 5, since pi^5 = e, and we had to raise it to the power of 5 to obtain the identity permutation.
(a) To write pi as a product of disjoint cycles, we start by considering each number individually.
For pi(1), we see that pi(1) = 1. Since pi(1) does not move, we can represent it as (1).
For pi(2), we see that pi(2) = 4. So, pi(2) moves to 4. pi(4) = 5, and pi(5) = 2. Since pi(2) goes to 4, pi(4) goes to 5, and pi(5) goes to 2, we can represent this as (2 4 5).
For pi(3), we see that pi(3) = 5. So, pi(3) moves to 5. pi(5) = 2, and pi(2) = 4. Since pi(3) goes to 5, pi(5) goes to 2, and pi(2) goes to 4, we can represent this as (3 5 2).
Putting it all together, we have pi = (1)(2 4 5)(3 5 2).
(b) To determine pi^2, we need to compose pi with itself.
(pi^2)(1) = pi(pi(1)) = pi(1) = 1.
(pi^2)(2) = pi(pi(2)) = pi(4) = 5.
(pi^2)(3) = pi(pi(3)) = pi(5) = 2.
(pi^2)(4) = pi(pi(4)) = pi(5) = 2.
(pi^2)(5) = pi(pi(5)) = pi(2) = 4.
Hence, pi^2 = (1)(5 2)(2 4) = (5 2)(2 4).
To determine pi^5, we can repeat the process of composing pi with itself four more times. However, note that pi^3 = (1)(2 4 5)(3 5 2) = (1)(2 4 5)(3 2 5) = (1)(2 5 3 4) = (2 5 3 4). Therefore, we have:
pi^4 = pi^3 * pi = (2 5 3 4) * (1 2 5 3 4) = (2 5 3 4)(1).
(pi^5)(1) = pi(pi^4(1)) = pi(2) = 4.
(pi^5)(2) = pi(pi^4(2)) = pi(5) = 2.
(pi^5)(3) = pi(pi^4(3)) = pi(3) = 5.
(pi^5)(4) = pi(pi^4(4)) = pi(4) = 1.
(pi^5)(5) = pi(pi^4(5)) = pi(1) = 1.
Therefore, pi^5 = (4 2)(5)(1)(3) = (4 2)(3).
To find pi^(-1), we need to find the inverse of pi. The inverse of a cycle is obtained by reversing the order of the elements in the cycle. Thus, pi^(-1) = (1)(2 5 3 4)(3 2 5) = (1)(4 3 5 2).
(c) To find the order of pi, we need to find the smallest positive integer k such that pi^k = (1).
Let's compute:
pi^2 = (5 2)(2 4) ≠ (1).
pi^3 = (5 2 4)(2 5 3) ≠ (1).
pi^4 = (2 5 3 4)(1) = (2 5 3 4) ≠ (1).
pi^5 = (4 2)(3) ≠ (1).
pi^6 = (4 2)(2 5 3)=(4 5 3)(1) = (4 5 3) ≠ (1).
pi^7 = (5 3)(2 4) ≠ (1).
pi^8 = (5 3 4)(2 5 3) ≠ (1).
pi^9 = (2 4)(1) = (2 4) ≠ (1).
From the calculations above, we see that pi^9 ≠ (1). Therefore, the order of pi is not 1. It's important to note that |pi| must divide the order of S(5), which is 5! = 120. Hence, we need to continue computing higher powers of pi to find its order.