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In Algebra 2e by Artin, under the group operations chapter, there is the following theorem:

Define a group operation of $G$ on a set $S$. Let $s\in S$ and $H=Stab(s)$. Then:

$$ \text{Suppose } as=s', s'\in S. \text{Then }\\ Stab(s') = aHa^{-1} = \lbrace g\in G : g=aha^{-1} \text{for } h\in H\rbrace$$

This means then that if two elements of $S$ are in the same orbit, then their stabilizers are conjugate subgroups. I guess what I'm looking for are examples where this is useful, and when it would be intuitive to utilize in a proof.

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  • $\begingroup$ Well, intuitively (or maybe geometrically), conjugation often corresponds to doing the same operation somewhere else. That is why it is useful in solving the Rubik's cube, because if you know how to do a $3$-cycle in one place, you can use conjugation to do it somewhere else. In your case, the operation in question is stabilizing a point in the set, and doing it somewhere else is stabilizing a different point, but it needs to be in the same orbit. $\endgroup$
    – Derek Holt
    Commented Apr 25, 2017 at 8:48

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