Exercise 1.14 (ii) from Rotman's book "An Introduction to the Theory of Groups". I'm trying to solve the exercise 1.14 (ii) from Rotman's book  "An Introduction  to the Theory if Groups" that says:
Let $\alpha$ and $\beta$ be cycles un $S_n$ (we  do not assume that they have the same  length). If there is $i_1$ moved by both $\alpha$ and $\beta$ and if $\alpha^k(i_1)=\beta^k(i_1)$ for all positive integers $k$, then $\alpha=\beta$
I don't know how to approach the exercise. Can someone help me with some hints?
 A: Let's break down the problem.
The section this problem appears in is about cycles, so what does it mean for $\alpha$ and $\beta$ to be cycles? It means we can denote them as $(j_1 \, j_2 \, j_3 \ldots \, j_r)$ and $(l_1 \, l_2 \, l_3 \ldots \, l_s)$.
What does it mean that $i_1$ is moved by $\alpha$ and $\beta$? That just means $\alpha(i_1)\neq i_1$, and $\beta(i_1)\neq i_1$, but in other words $i_1=j_p$ for some $p$ and $i_1=l_q$ for some $q$; that is, the character (if we are thinking of $S_n$ as the group of permutations of $n$ characters) appears in these cycles.
To make things concrete, consider $S_4$ where we let the characters be $\{1,2,3,4\}$. Then the permutation $(123)$ is the one that takes $1$ to $2$, $2$ to $3$ and $3$ to $1$, and fixes $4$. This permutation is a $3$-cycle, where three denotes its length. We can also write the permutation as $(123)(4)$, to emphasize $4$ is fixed, but its not necessary. This permutation moves the characters $1, 2,$ and $3$. 
Back to the general case, we have that $\alpha^k(i_1)=\beta^k(i_1)$ for all positive integers $k$. This means that after one application of $\alpha$ or $\beta$, $i_1$ gets sent to the same thing, after two repeated applications of $\alpha$ or $\beta$, $i_1$ gets sent to the same thing, etc. 
I hope this helps!
A: My suggestion is that you write some cycle, fix a point $i_1$, and look at what the powers $\alpha^k(i_1)$ do. You should see the pattern very quickly. 
