On the limit distribution of the number of cycles and the logarithm of the order of a class of permutations

Let $S_n$ be the symmetric group of degree $n$ and let $S_n^{(k)}$ be the set of permutations $a\in S_n$ such that the equation $x^k=a$ has a solution $x\in S_n$. Consider the uniform probability distribution on the set $S_n^{(k)}$.
This article investigates the limit distributions on $S_n^{(k)}$, as $n\to\infty$ and for fixed $k\geqslant2$, of the random variables $\xi_s$, $\eta$, and $\zeta$, where $\xi_s$ is the number of cycles of length $s$, $\eta$ is the number of all cycles, and $\zeta$ is the logarithm of the order of a random permutation $a\in S_n^{(k)}$.
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