I posted 3 full questions to give context. But my main problem is the second part of the questions and how would they be answered/proved.
2026-04-04 13:02:51.1775307771
Orbit-Stabilizer Theorem proofs?
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If you want to show that two sets have the same number of elements, you can try to construct a bijection between them.
Explicitly, if you have $y = gx$, you can construct a bijection between $Stab_G(x)$ and $Stab_G(y)$ as follows: If $h \in Stab_G(x)$, that means $hx = x$. I claim that $ghg^{-1} \in Stab_G(y)$. Furthermore, if $k \in Stab_G(y)$, $g^{-1}kg \in Stab_G(x)$. (Try to prove these two assertions). So I have constructed a pair of functions which are each other's inverse, and so they are both bijective.
As for the other question, is depends what 'formula for the size of an orbit' they're using. If it is $|O(x)| = |G|/|Stab_G(x)|$, then it's clear from that formula that the size must divide the order of $G$.