The orbit-stabilizer theorem implies, very immediately, one of the most important counting results in group theory. The proof is easy enough to give in a paragraph now that we’ve set up the requisite machinery. Remember that we counted fixed points by looking at the size of the stabilizer subgroup. Let’s count them another way. Since a fixed point is really a pair such that , and we’ve been counting them indexed by , let’s count them indexed by . We use to denote the set of fixed points of . (Note that this is a function of the group action, not the group, but again we’re abusing notation.) Counting the total number of fixed points “vertically,” then “horizontally,” gives the following.

**Proposition:** .

On the other hand, by the orbit-stabilizer theorem, it’s true for any orbit that , since the cosets of any stabilizer subgroup partition . This immediately gives us the lemma formerly known as Burnside’s, or the Cauchy-Frobenius lemma, which we’ll give a neutral name.

**Orbit-counting lemma:** The number of orbits in a group action is given by , i.e. the average number of fixed points.

In this post we’ll investigate some consequences of this result.

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