maximum order of an element in symmetric group

Question

While doing my homework i find out that the maximum order of an element in $S_3$ is 3 (the element $(123)$) and the maximum order of an element in $S_4$ is 4 (the element $(1234)$)

Can i generalize that and say that the maximum order of an element in $S_n$ is n?

Answer 1

Well after checking $S_5$ i saw that what i said is not correct: element $(123)(45)$ has order of 6. thanks @vadim123

Answer 2

Your answer is here.

The maximum order of an element of finite symmetric group by William Miller, American mathematical monthly, page 497-506.

Answer 3

In general, you have to look at disjoint cycle types and hence for a "maximum" partition of $n=a_1 + \dots a_k$ and take the lcm of the $a_i$'s. Example for $S_7$: $7=2+5$ gives you order $10$ but $7=3+4$ yields $12$.