## Find all possible orders of elements in Sym(7)

Find all numbers $n$ such that $S_7$ contains an element of order $n$.

Note that the only possible cycle shapes of elements in $S_7$ are $1$, $(2)$, $(2,2)$, $(2,2,2)$, $(3)$, $(3,2)$, $(3,2,2)$ $(3,3)$, $(4)$, $(4,2)$, $(4,3)$, $(5)$, $(5,2)$, $(6)$, and $(7)$. By a previous exercise, the order of a product of disjoint cycles is the least common multiple of the lengths of the factors. So the possible orders of elements in $S_7$ are (in increasing order) 1, 2, 3, 4, 5, 6, 7, 10, and 12. $\blacksquare$