show/hide this revision's text 3 cleaned up a bit, emphasized that "average theorem" was main idea; added 14 characters in body

Burnside's Lemma states that, given a set $X$ acted on by a group $G$,

$$|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|$$

where $|X/G|$ is the number of orbits of the action, and $|X^g|$ is the number of fixed points of $g$. In other words, the number of orbits is equal to the average number of fixed points of an element of $G$.

Is there any way in which the fixed points of an element $g$ can be thought of as orbits? I imagine the result might be somewhat more intuitive had wondered aloud on my recent question here how (or ifwe could interpret it ) Burnside's Lemma can be interpreted as having the same kind of object on both sides, so as to be a "true" average theorem, e.g.

"the number of orbits = the average over $g\in G$ of (number of orbits satisfying {(something to do with $g$}"g$))"

or

"the number of orbits = the average over $g\in G$ of {the (number of orbits of some new action which depends on $g$}"g$)"

Since Qiaochu recently stated in the comments hereto my question that he suspects Burnside's Lemma can be categorified, and that this may be related, I have also added that tag.
show/hide this revision's text 2 deleted 36 characters in body; deleted 2 characters in body

Burnside's Lemma states that, given a set $X$ acted on by a group $G$,

$$|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|$$

where $|X/G|$ is the number of orbits of the action, and $|X^g|$ is the number of fixed points of $g$. In other words, the number of orbits is equal to the average number of fixed points of an element of $G$.

Perhaps this is silly or naive, but is

Is there any way in which the fixed points of an element $g$ can be thought of as "orbits"? orbits? I imagine the result might be somewhat more intuitive if we could interpret it as having the same kind of object on both sides, e.g.

"the number of orbits = the average number of orbits satisfying {something to do with $g$}"

or

"the number of orbits = the average of {the number of orbits of some new action which depends on $g$}"

Since Qiaochu recently stated in the comments here that he suspects Burnside's Lemma can be categorified, and that this may be related, I have also added that tag.
show/hide this revision's text 1

Intuitive explanation of Burnside's Lemma

Burnside's Lemma states that, given a set $X$ acted on by a group $G$,

$$|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|$$

where $|X/G|$ is the number of orbits of the action, and $|X^g|$ is the number of fixed points of $g$. In other words, the number of orbits is equal to the average number of fixed points of an element of $G$.

Perhaps this is silly or naive, but is there any way in which the fixed points of an element $g$ can be thought of as "orbits"? I imagine the result might be somewhat more intuitive if we could interpret it as having the same kind of object on both sides, e.g.

"the number of orbits = the average number of orbits satisfying {something to do with $g$}"

or

"the number of orbits = the average of {the number of orbits of some new action which depends on $g$}"

Since Qiaochu recently stated in the comments here that he suspects Burnside's Lemma can be categorified, and that this may be related, I have also added that tag.