On the cardinality of a factor set in the symmetric group

TL;DR

This paper investigates the properties of a specific equivalence relation in the symmetric group generated by a cycle, and introduces a graph-based algorithm to count the number of equivalence classes.

Contribution

It defines a new equivalence relation in symmetric groups and develops a graph-based algorithm to determine the number of classes under this relation.

Findings

Defined a $\sigma$-equivalence relation in $ ext{Sym}_n$

Constructed a finite oriented graph $\Gamma_n$ to analyze the relation

Provided an algorithm to count equivalence classes

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.