On the cardinality of a factor set in the symmetric group

الأصل · EN

Let n be a positive integer, σ be an element of the symmetric group Sₙ and let σ be a cycle of length n. The elements α,β∈ Sₙ are σ-equivalent, if there are natural numbers k and l, such that σᵏ α=βσˡ, which is the same as the condition to exist natural numbers k₁ and l₁, such that α= σᵏ¹ βσˡ¹. In this work we examine some properties of the so defined equivalence relation. We build a finite oriented graph Γₙ 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.