Exercise 9, § 5

Let G be a group of permutations of a finite set E; for all $s \in G$, let $\chi(s)$ denote the cardinal of the set of fixed points of s. (a) Show that $\sum_{s \in G} \chi(s) = N t$, where N = Card(G) and t is the number of orbits of G in E (evaluate in two ways the number of ordered pairs $(s, x) \in G \times E$ such that $s(x) = x$). (b) Suppose that, for all $a \in E$, the stabilizer $H_a$ of a is not reduced to e. Show that, if $\chi(s)$ is independent of s for $s \neq e$ and is equal to an integer k, then

(Note that Card(E) $\leq Nk$.)

In the particular case where $k = 2$, find the possible orders of the groups $H_a$; show that, if $t = 3$, the order of $H_a$ cannot be $\geq 3$ for elements of two of the three orbits unless N has one of the values 12, 24 or 60.

(c) Suppose that G operates transitively on E and let H denote the stabilizer of an element of E; show that $\sum_{s \in G} \chi(s)^2$ is equal to N \cdot t_H, where $t_H$ is the number of orbits of H in E. (For every $(s, u)$, write $\chi(s, u) = \chi(s)$. Evaluate in two ways the sum $\sum_{(s, u) \in R} \chi(s, u)$, where R is the set of ordered pairs $(s, u)$ such that $usu^{-1} \in H$.)