Discussion
In the permutation group \(S_{6}\), the number of elements of order 8 is
Answer: A
There are 0 elements of order 8.
The order of a permutation is the least common multiple of the lengths of the cycles in its disjoint cycle decomposition. In \(S_{6}\) the cycle lengths add up to at most 6.
For the order to be 8, some cycle length would have to be divisible by 8 (an lcm of numbers each at most 6 is divisible by 8 only if one of them is), which needs a cycle of length at least 8. That is impossible with 6 points. The largest power of 2 that occurs as an order in \(S_{6}\) is 4, from a 4-cycle.
So the counts 1, 2 and 4 are all wrong; the smallest symmetric group with an element of order 8 is \(S_{8}\).
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