If $\beta$ is the transposition $\beta = (1, 4)$, compute both $\beta\alpha$ and $\beta\alpha\beta^{−1}$ and compute orders. Let $\alpha$ be the permutation in the symmetric group $S_9$ defined by
$$\alpha =\left(\begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 5 & 1 & 7 & 6 & 3 & 9 & 2 & 8 & 4\end{matrix}\right)$$

If $\beta$ is the transposition $\beta = (1, 4)$, compute both $\beta\alpha$ and $\beta\alpha\beta^{−1}$ and give a list of those positive integers which are the orders of the elements in the
  symmetric group $S_9$. For each such integer, also give one permutation having that
  order.

So I have found out that $\alpha = (1,5,3,7,2)(4,6,9)$ as a product of two disjoint cycles. However, I am not sure how to use the transposition of $\beta$ in this case to compute the composition of $\beta\alpha$ and $\beta\alpha\beta^{−1}$. 
Also for the second part of the question, wouldn't the orders of the elements in the first cycle $(1,5,3,7,2)$ all be 5 since thats how many elements it takes to cycle? and similarly for the second cycle $(4,6,9)$ they would all be order 3? 
I know this is a few few parts but I'm having some clarification issues, thank you for any and all help.

In the symmetric group $S_n$, the group operation is the composition of permutations. The permutation $\beta\alpha$ is the permutation $\alpha$ followed by $\beta$. For example, $(\beta\alpha)(1)=\beta(\alpha(1))=\beta(5)=5$.
To answer your second question, $1,2,...n$ are not elements of $S_n$. The elements are $\textit{permutations}$ on $1,2,...n$.
You are correct that the order of a permutation that is a single cycle is the length of that cycle. More generally, if a permutation is a product of multiple disjoint cycles, its order is the least common multiply of the lengths of those cycles.
