Question

The number of elements of order 12 in the symmetric group S₇ is equal to____

Answer 1

Correct Answer: 420

The symmetric group \(S_7\) consists of all permutations of 7 elements. To find the number of elements of order 12 in \(S_7\), we must consider the cycle structure of these permutations. A permutation in \(S_7\) being of order 12 means the least common multiple (LCM) of the cycle lengths in its disjoint cycle representation is 12. To achieve this, the cycle types we consider that fulfill LCM requirements are:

• (12): Not possible in \(S_7\).
• (6,2): Also, not possible in \(S_7\).
• (4,3): This is feasible. Here's why and how we find it.

For a permutation to have a cycle structure \((4,3)\), it must include a 4-cycle and a 3-cycle. Let's compute the possibilities:

• Select 4 elements out of 7 for the 4-cycle: \( \binom{7}{4} = 35\).
• There are \((4-1)!=3!\) ways to arrange these elements in a 4-cycle, equal to 6 ways.
• Choose the remaining 3 elements for the 3-cycle: \(1\) way since they are the only elements left.
• The number of 3-cycles formed is \((3-1)!=2\).

 

Total number of permutations = \( 35 \times 6 \times 2=420\).

Thus, the number of elements in \(S_7\) with order 12 equals 420. 

The final result: 420.