QUESTION
For a permutation p: X → X, let $p^{k}$ denote the permutation arising by a k-fold composition of p, i.e. $p^{1} = p \ and \ p^{k} = p◦p^{k−1}$. Define a relation ≈ on the set X as follows: i ≈ j if and only if there exists a k ≥ 1 such that $p^{k}(i) = j$. Prove that ≈ is an equivalence relation on X, and that its classes are the cycles of p.
I am new to this, what does it mean by its classes? does it mean its elements in the set X? I do see the connection between the k-fold composition and the permutation of p. I think I just do not understand the language. Please give me some hints.
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