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## Homework Statement

If [tex]\alpha,\beta\in S_n[/tex] and if [tex]\alpha \beta = \beta \alpha[/tex], prove that [tex]\beta[/tex] permutes those integers which are left fixed by [tex]\alpha[/tex]. Show that [tex]\beta[/tex] must be a power of [tex]\alpha[/tex] when [tex]\alpha[/tex] is a n-cycle.

The other way round is easy to see, since if two cycles are disjoint they do not do anything with the numbers permuted by the other cycle, hence they commute. But I don't know how to start when I want to prove the statement above... can anyone hint me?