I couldn't find how I should go to the result in the following problem. ( Problem 5.17, Isaac's Character Theory Book )
Let $H \leq G$ and let $\chi = (1_H)^{G}$. Fix a positive integer $n$. For $g \in G$, let
$m(g)= \langle \;\chi_{<g>} ,1_{<g>} \; \rangle$ and define $\theta(g) = n^{m(g)}$. Show that $\theta$ is a character of $G.$
My guess is that we need to find an $\mathbb{C}G-$module such that the representation afforded by this module affords that character $\theta(g) = n^{m(g)}$. Actually Isaac gave a hint for it, but I couldn't understand it. Could you help ?
Hint: Let $\Omega$ be the set of right cosets of $H$ in $G$. Fix a set $S$ with cardinality n
and let $\lambda$ be the set of all functions $\Omega \rightarrow S$ . Then $G$ acts on $\lambda$.
I think the action of $G$ "$f.g$" should be the function on $\Omega$ such that $f.g (Hx) = f(Hxg)$ for any $f$ in $\lambda$. What we can say about the representation afforded by the module formed by this group action ?
Best Answer
The set $\Lambda$ of all functions $f:H \backslash G \to S$ is a $G$-set with action defined by $$(g \cdot f)(Hx)=f(Hxg).$$ The permutation representation afforded by this $G$-set has, like any permutation representation, character $\eta$ given by the number of fixed points $$\eta(g)=|\{f \in \Lambda \ | \ g \cdot f=f \}|.$$ Now $f$ is fixed by $g$ exactly if it is constant on $g$-orbits on $H \backslash G$, so the number of $f$ fixed by $g$ is $n^{m_g}$, where $m_g$ is the number of $g$-orbits in $H \backslash G$. To complete the proof we must therefore prove that $m_g=m(g)$ with your notation.
The character $\chi$ is the character of the permutation representation of $G$ on $H \backslash G$. Thus $\chi(x)$ is the number of right cosets fixed by $x$. Hence writing $k$ for the order of $g$ we have $$m(g)=\langle \chi_{\langle g \rangle},1_{\langle g \rangle} \rangle=\frac{1}{k} \sum_{j=0}^{k-1} \chi(g^j).$$ This is the average number of fixed points of elements of $\langle g \rangle$, which by Burnside's counting theorem is the number of orbits $m_g$ of $g$ on $H \backslash G$.