A proof of Burnside's lemma first claims the number of equivalence classes on G, the given set of permutations, $\sigma$'s, to be applied to an object (i.e. a beaded necklace to be colored), is equal to
$\sum_{k\in K}\frac{1}{|O_k|}$. (1)
$k$ is a certain coloring of the object (e.g. a 4-bead necklace with k=bbww means first two are black and last two are white) in the set $K$ of all colorings, and $O_k$ is the orbit set of that $k$.
From here, the proof lets $O_{k_m}=\{k_1, k_2, ...,k_m\}$ for some $k_m$. (2)
$\Rightarrow|O_{k_m}|=m$ ($m$ is the number of 'colors' available; $C=\{$white$, $ black$\}$, $m=2$).
$\Rightarrow O_{k_1}=O_{k_2}=...=O_{k_m}$. (3)
The proof goes on to show that the presented Burnside's Lemma, $\frac{1}{|G|}\sum_{\sigma\in G}|$Inv$(\sigma)|=\sum_{k\in K}\frac{1}{|O_k|}$, where Inv($\sigma$) is the invariant set of $\sigma$.
My questions are about (1),(2),(3):
How was (1) claimed right off the bat? Is there an intuitive reason?
(2) seems to imply that there is a coloring $k_m$ that can become any other coloring via some $\sigma$. Wouldn't this mean that there is only one equivalence class? (3) seems to show this as well. Clearly, there are situations where there is more than one equivalence class..
I don't understand how they are claiming these crucial steps for the proof.
EDIT: Original can be found next to 'download' near the top at http://ntnu.diva-portal.org/smash/record.jsf?pid=diva2:348745. Proof is on page 12 of the pdf.