Let σ∈sx. if σn(x)=y, we will say that x∼y. i. show that ∼ is an equivalence relation on x. ii. if σ∈an and τ∈sn, show that τ−1στ∈an.τ−1iii. define the orbit of x∈x under σ∈sx to be the setox,σ={y: x∼y}.compute the orbits of each of the following elements in s5: α=(1254)β=(123)(45)γ=(13)(25).iv. if ox,σ∩oy,σ≠∅,ox, prove that ox,σ=oy,σ.the orbits under a permutation σ are the equivalence classes corresponding to the equivalence relation ∼.v. a subgroup h of sx is transitive if for every x, y∈x, there exists a σ∈h such that σ(x)=y. prove that 〈σ〉 is transitive if and only if ox,σ=x for some x∈x(abstract algebra)