How Many Elements Have Order 12 in the Symmetric Group S7?

Описание: In the Symmetric Group on seven objects S7, how many elements have order 12? When a permutation in S7 is written in disjoint cycle notation, the order of the permutation is the least common multiple (LCM) of the lengths of the cycles. The only way to use the seven objects {1,2,3,4,5,6,7} once in such cycles and get an LCM equal to 12 is if it the permutation is written as (a1 a2 a3)(a4 a5 a6 a7) (where the a's are all distinct so that the cycles are disjoint), which can also be written as (a4 a5 a6 a7)(a1 a2 a3) since disjoint cycles commute. There are 7!/(3*4) = 420 such permutations (note also that, for example, (a1 a2 a3) = (a2 a3 a1) = (a3 a1 a2)). Overall, S7 has 7! = 5040 elements (the order of S7 is |S7| = 7! = 5040).

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