In section 4.1 Group Actions and Permutation Representations Dummit and Foote define an orbit as follows:

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"Let G be a group acting on the nonempty set A.

The equivalence class {g.a | g $\displaystyle \in$ G} is called the orbit of G containing a."

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However, Fraleigh [A First Course in Abstract Algebra] defines Orbits in Section 9: Orbits, Cycles and Alternative Groups as follows: [calling it an orbit of a permutation]

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"Each permutation $\displaystyle \sigma$ of a set A determines a natural partition of A into cells with the property that a,b $\displaystyle \in$ A are in the same cell if an only if b = $\displaystyle \sigma^n(a)$ for some n$\displaystyle \in$ Z"

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Are these two definitions equivalent. If so, can someone please explicity & formally demonstrate how/why they are equivalent?

Peter