Exercise 13, § 21

Let $\mathscr{H}$ be a set of subgroups of G.

a) Prove that the following properties are equivalent:

(i) The union of the conjugates of the subgroups belonging to $\mathscr{H}$ is equal to G.

(ii) Every character of G is a linear combination with rational coefficients of characters induced by the characters of subgroups belonging to $\mathscr{H}$.

(Deduce from Proposition 6 of VIII, p. 412 that property (ii) is equivalent to the injectivity of the restriction homomorphism $\mathscr{Z}_K(G)\rightarrow \bigoplus_{H\in\mathscr{H}}\mathscr{Z}_K(H).$)

b) Let $\mathscr{C}$ be the set of cyclic subgroups of G; it obviously has property (i). For every cyclic subgroup C of G, we denote by $\varphi_C$ the function on C with value Card(C) on the generators of C and 0 on the other elements. Prove the equality Card(G) $=\sum_{C\in\mathscr{C}}$ Ind$^G_C(\varphi_C)$.

c) Let $C\in \mathscr{C}$. Prove that the function $\varphi_C$ belongs to the subgroup $R_K(G)$ of $\mathscr{Z}_K(G)$ generated by the characters (reason by induction on the order of C, using b)).

d) Let $\chi$ be the character of an irreducible representation G. Prove the equality $\chi =$ Card(G)$^{-1}\sum_{C\in\mathscr{C}}$ Ind$^G_C(\varphi_C$ Res$^G_C(\chi ))$, which gives an explicit formula for property (ii) of a).