Let $p$ be a permutation, and let $p^k$ denote the permutation arising by a k-fold composition of $p$. By the order of the permutation $p$ we mean the smallest natural number $k$ ≥ 1 such that $p^k = id$, where $id$ denotes the identity permutation (mapping each element onto itself). Show that each permutation $p$ of a finite set has a well-defined finite order, and show how to compute the order using the lengths of the cycles of $p$.
This is my first proof for a combinatorics class, and so I'm struggling to figure out how to actually write the proof. I did some examples, and I quickly figured out that the order $k$ of a permutation $p$ is the $LCM(L_1,L_2,\cdots,L_n)$ if we let $L_n$ equal the length of the nth cycle. If I prove that, then I think I answered the question because I (1) show how to compute the order and (2) show it's well-defined because there's only one LCM. But I just don't know how to prove that english statement, nor do I know how to translate that english sentence into mathematical language.