Question

The maximum order of a permutation \( \sigma \) in the symmetric group \( S_{10} \) is ................

Hint:

The order of a permutation is the LCM of the lengths of its disjoint cycles. To maximize the order, the cycle lengths should be as large as possible.

Answer 1

Correct Answer: 29.9 - 30.1

Step 1: Understanding the structure of the symmetric group.
The symmetric group \( S_{10} \) consists of all the permutations of 10 elements. The order of a permutation in \( S_n \) is the least common multiple (LCM) of the lengths of the disjoint cycles in its cycle decomposition.
Step 2: Finding the maximum order of a permutation.
To maximize the order of a permutation in \( S_{10} \), we want to decompose the permutation into the maximum number of disjoint cycles. The maximum order is achieved by choosing the longest possible disjoint cycles. The largest possible permutation in \( S_{10} \) is one where the cycle decomposition is as follows: \[ (1 \, 2 \, 3 \, 4 \, 5) (6 \, 7 \, 8) (9 \, 10) \] The order of this permutation is the LCM of the lengths of the cycles: \[ \text{LCM}(5, 2, 2) = 10 \]
Step 3: Conclusion.
Thus, the maximum order of a permutation in \( S_{10} \) is \( \boxed{30} \).