# Problem 5.17, Isaac's Character Theory Of Finite Groups

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 $$m(g)= \langle \;\chi_{} ,1_{} \; \rangle$$ 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 ?

## 1 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 $$H \backslash G$$.