I have an exercise scribbled down, and I am not sure what it is asking. It is somewhat similar to Burnside's lemma.
We have a finite group $G$ acting on a set $X$. For each $g \in G$, let $X^g$ denote the set of elements in $X$ fixed by $g$.
$\sum_g |X^g|^2 = |G| \cdot \text{(number of orbits of a stabilizer)}$
I am not sure what it means by "orbit of a stabilizer". I am guessing that it refers to the action of $G$ on cosets of a stabilizer by multiplication. But this really doesn't make sense to me since this action is transitive and the orbit is just the entire set.
Does anyone know of such an exercise and can someone explain what the precise statement of the problem should be?