It is reasonably well-known that the variance of the number of fixed points for $S\_n$ equals $1.$ Now, what about other transitive permutation groups on $\{1, \dotsc, n\}?$ Presumably much is known. I am particularly curious about which groups have the highest variance...



variance of the number of fixed points for a permutation group
Let $\pi(g)$ be the number of fixed points of a permutation $g$ on $\Omega=\{1,2,\dots,n\}$, and $G$ be a transitive permutation group on $\Omega$. Then, by the orbit counting formula (usually attributed to Burnside, but previously known by Cauchy and Frobenius too), the variance $V(\pi)=E(\pi^2)-E(\pi)^2=E(\pi^2)-1=\frac{1}{|G|}\sum\_{g\in G}\pi(g)^2-1$ is $r-1$, where $r$ is the number of orbits of $G$ on $\Omega\times\Omega$. Note that $r$ is also the number of orbits of a point stabilizer $G\_1$ on $\Omega$. In particular, $V(\pi)\le n-1$, with equality if and only if $|G|=|\Omega|=n$. 


