For n ≥ 1, let $S_n$ denote the group of all permutations on n symbols. Which of the following statements is true ?
A. S3 has an element of order 4
B. S4 has an element of order 6
C. S4 has an element of order 5
D. S5 has an element of order 6.
My work: Order of group is $n!$.
A and C are false (by Lagrange theorem). Now how can i proceed next intuitively ?
Question B
An element of $\mathcal S_4$ is either a $2$-cycle (of order $2$), a $3$-cycle (having order $3$), a product of two $2$-cycles (having order $2$) or a $4$-cycle (of order $4$). Hence none of the elements of $\mathcal S_4$ may have order $6$.
Question D
A product of two disjoint cycles of order $2$ and $3$ has indeed order $6$ in $\mathcal S_5$. This is the case for example of $(1 \ 2 \ 3)(4 \ 5)$.