This article is cited in 3 papers

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

A. I. Pavlov


Abstract: 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.

UDC: 519.21

MSC: 05A15, 20B30

Received: 20.11.1979

 English version: Mathematics of the USSR-Sbornik, 1981, 40:3, 349–362

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