Let x be the 100-cycle (1 2 3 100) and let y be the transposition (49 50) in the permutation group S100. Then the order of xy is ______Correct answer is between '99,99'. Can you explain this answer?

Question

Let x be the 100-cycle (1 2 3  100) and let y be the transposition (49 50) in the permutation group S100. Then the order of xy is ______Correct answer is between '99,99'. Can you explain this answer?

Answer

99,99

Answer

Given information:
We are given two permutations in the permutation group S100:
- x = (1 2 3 ... 100) (a 100-cycle)
- y = (49 50) (a transposition)

To find:
We need to determine the order of the permutation xy.

Solution:
1. Order of a permutation:
The order of a permutation is the smallest positive integer k such that raising the permutation to the power of k gives the identity permutation.
In other words, if p is a permutation and p^k = e, where e is the identity permutation, then the order of p is k.

2. Order of a product of permutations:
The order of a product of permutations is the least common multiple (LCM) of their individual orders.
If the order of permutation p is m and the order of permutation q is n, then the order of the product pq is LCM(m, n).

3. Order of the 100-cycle:
The order of the 100-cycle (1 2 3 ... 100) is 100.
This can be seen by observing that raising the 100-cycle to the power of 100 gives back the original permutation.

4. Order of the transposition:
The order of a transposition (a b), where a and b are distinct integers, is 2.
This can be seen by observing that raising the transposition to the power of 2 gives back the identity permutation.

5. Order of the product xy:
To find the order of the product xy, we need to find the LCM of the orders of x and y.

- The order of x is 100.
- The order of y is 2.

The LCM(100, 2) is 100, which means raising the product xy to the power of 100 will give back the identity permutation.

Conclusion:
The order of the permutation xy is 100, which means the correct answer is between '99,99'.

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