Here is the exercise from a Pinter's "A book of abstract algebra" from a chapter dealing with permutations on a finite set:
Let $\alpha$ and $\beta$ be cycles, not neccessarily disjoint. Prove that, if $\alpha^2=\beta^2$, then $\alpha=\beta$.
I think I've found counterexample: $(2345)^2 = (24)(35) = (2543)^2$. Am I right?