Suppose that τ in Sn fixes no symbol. Show that τ = μ^m for some n-cycle μ and positive integer m if and only if τ is the product of disjoint cycles of equal length.
I know that τ can be written as the product of disjoint cycles, but am not sure how to proceed from there.
See attached file for full problem description.
Symmetric groups and disjoint cycles are investigated.