Question

Which one of the following is TRUE for the symmetric group \( S_{13} \)?

• A. \( S_{13} \) has an element of order 42
• B. \( S_{13} \) has no element of order 35
• C. \( S_{13} \) has an element of order 27
• D. \( S_{13} \) has no element of order 60

Hint:

The order of an element in \( S_n \) is the least common multiple of the lengths of the disjoint cycles in its cycle decomposition.

Answer 1

The Correct Option is A

The order of an element in the symmetric group \( S_n \) corresponds to the least common multiple (LCM) of the lengths of the disjoint cycles in its cycle decomposition. We are asked to determine if the symmetric group \( S_{13} \) has an element of order 42. We can find the LCM of the cycle lengths. For example, we can have an element that is a product of a 6-cycle and a 7-cycle, since the LCM of 6 and 7 is 42. Hence, \( S_{13} \) does indeed have an element of order 42. - \( S_{13} \) has no element of order 35, because there is no valid way to combine cycles of lengths that result in an LCM of 35. - \( S_{13} \) does not have an element of order 27, because 27 is not the LCM of any combination of integers less than or equal to 13. - \( S_{13} \) does have an element of order 60, because the LCM of the cycles (5, 3, and 4) is 60. Hence, option (D) is false. Thus, the correct answer is (A) \( S_{13} \) has an element of order 42.