Question

Let \( x \) be the 100-cycle \( (1 \, 2 \, 3 \, \dots \, 100) \) and let \( y \) be the transposition \( (49 \, 50) \) in the permutation group \( S_{100} \). Then the order of \( xy \) is

Hint:

The order of a product of a cycle and a transposition is the least common multiple of the lengths of the involved cycles.

Answer 1

Correct Answer: 99

Step 1: Understanding the cycle structure.
The permutation \( x \) is a 100-cycle, meaning it cycles through 100 elements. The permutation \( y \) is a transposition, swapping two elements (49 and 50).
Step 2: Analyze the product \( xy \).
When you multiply the two permutations \( x \) and \( y \), the action of \( y \) on the cycle \( x \) affects only the two elements it swaps, while the other elements remain unaffected by \( y \). The order of the product of a cycle and a transposition is the least common multiple (LCM) of the lengths of the cycles involved.
Step 3: Determine the order of \( xy \).
Since \( x \) is a 100-cycle and \( y \) swaps just two elements, the order of \( xy \) is 50, because the product \( xy \) will divide the 100-cycle into two disjoint cycles of length 50. Hence, the order of \( xy \) is 50.
Step 4: Conclusion.
Thus, the order of \( xy \) is \( \boxed{50} \).