On the number of solutions of the equation $x^k=a$ in the symmetric group $S_n$

This paper consists of three sections. In the first a formula is given for the number $N^{(k)}_n(a)$ of solutions of the equation $x^k=a$ in $S_n$ depending on the cyclic structure of the permutation $a$. In the second an asymptotic formula is given for the quantity $M^{(k)}_n=\max_{a\in S_n}N^{(k)}_n(a)$ for a fixed $k\geqslant2$ as $n\to\infty$. In the third an asymptotic formula is found for the cardinality of the set of permutations $a$ such that the equation $x^k=a$ has a unique solution.
Bibliography: 5 titles.