On the cardinality of a factor set in the symmetric group

Abstract

Let $n$ be a positive integer, $\sigma$ be an element of the symmetric group ${\mathcal{S}}_n$ and let $\sigma$ be a cycle of length $n$. The elements $\alpha ,\beta \in {\mathcal{S}}_n$ are $\sigma$-equivalent, if there are natural numbers $k$ and $l$, such that ${\sigma }^k\alpha =\beta {\sigma }^l$, which is the same as the condition to exist natural numbers $k_1$ and $l_1$, such that $\alpha ={\sigma }^{k_1}\beta {\sigma }^{l_1}$. In this work we examine some properties of the so defined equivalence relation. We build a finite oriented graph ${\Gamma }_n$ with the help of which is described an algorithm for solving the combinatorial problem for finding the number of equivalence classes according to this relation.