Exercise 12, § 21

Let H be a subgroup of G; suppose that for every $g\in G$- H, we have the equality $H\cap gHg^{-1}=\{1\}$.

a) Let $u$ be a central function on H. Prove that the function Ind$^G_H(u)$ coincides with $u$ on H $-\{1\}$. If $v$ is another central function on H such that $v(1) = 0$, then we have $\langle$Ind$^G_H(u)$, Ind$^G_H(v)\rangle_G=\langle u, v\rangle_H$.

b) Let $\lambda \in \widehat{H}$; we denote by $\chi '_{\lambda}$ the function $\chi_{\lambda}-d_{\lambda}$ on H and by $\theta_{\lambda}$ the function $d_{\lambda}+$ Ind$^G_H(\chi '_{\lambda})$ on G. Prove that $\theta_{\lambda}$ is the character of an irreducible representation of G of degree $d_{\lambda}$ whose restriction to H is isomorphic to $\lambda$ (use a) to calculate $\langle \theta_{\lambda}, \theta_{\lambda}\rangle ,\langle \theta_{\lambda},1\rangle$, and $\langle$Res$^G_H(\theta_{\lambda}), \chi_{\lambda}\rangle )$.

c) Set $\theta =\sum_{\lambda}d_{\lambda}\theta_{\lambda}$. Prove that we have $\theta (h) = 0$ if $h\in H-\{1\}$ and $\theta (g) =$ $\theta (1) =$ Card(H) if $g$ does not belong to any conjugates of H.

d) Let $L'$ be the set of elements of G that do not belong to any conjugates of H, and set $L = L'\cup \{1\}$. Prove that L is a normal subgroup of G (deduce from c) that L is the kernel of a representation with character $\theta )$.

e) Prove that G is the semidirect product of H and L (calculate the order of L). f ) Let $h\in H-\{1\}$. Prove that the automorphism int($h$) restricted to L has no fixed point other than 1. Deduce that the order of H divides Card(L) $-1$.

g) Prove that if the subgroup H has even order, then L is commutative (apply Exercise 23, d) of I, §6, p. 145).