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I don't understand yet:in order to find orbits of a given permutation of a set $A$, is it necessary that the relation $\sim$ involving elements $a$ and $b$ to be an equivalence relation? Nice day.

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    if you mean the relation $a\sim b$ iff $\exists g\in G : ga = b$ then yes this is an equivalence relation, and the orbits are precisely the equivalence classes.2011-12-20
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    It is possible that neema means with orbit an equivalence class of an equivalence relation given by the cycles of the permutation. For example, Herstein does this in Topics in Algebra ($2$nd ed, pg. $77$). Fix some permutation $g \in S_A$. An orbit is an equivalence class of the equivalence relation $\sim$ on $A$, where $a \sim b$ if and only if $a = g^i(b)$ for some integer $i$. But I'm still confused by this question.2011-12-20

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