I'm reviewing for a midterm, and one of the questions is:

"What is the smallest such n so $\displaystyle S_n$ contains an element of order 30? Give reasons."

I don't know how to solve this. Clearly the order of the group must be at least 30, so n is greater than 4. Also 30 must divide the order of the group, by Lagrange's Theorem, but it does for every n greater than 4.

Thanks for any help!