# The Riemann-Hurwiz formula and a special type of group action

Suppose $G$ is some finite group acting on the set $A$, with the special condition that each element of $G$ fixes $n$ points of $A$. Then we can write $n(|G|-1) = \sum_{a \in A}{(S_G(a)-1)$. (Counting the non-identity elements of $G$ in two different ways.)
Now this formula strongly resembles the Riemann-Hurwiz formula, which relates the Euler characteristics of two Riemann surfaces $M_1\to M_2$, one of which is a degree $n$ ramified covering of the other : we then have have $\chi(M_1)=N\chi(M_2)-\sum_{p \in M_1} (r_{p} -1)$, where $r_p$ is the ramification index at $p$.