Exercise 27, § 21

Let F be an algebraically closed field of characteristic $p >0$ and G be a finite group.

a) Let $x\in G$. Prove that there exists a unique decomposition $x=x_ux_s$, where the order of $x_u$ is a power of $p$, that of $x_s$ is prime to $p$, and $x_u$ and $x_s$ commute. The elements $x_u$ and $x_s$ are powers of $x$.

b) Let $\pi$ be a linear representation of G over F of finite degree. Prove that we have $\chi_{\pi}(x) =\chi_{\pi}(x_s)$ for every $x\in G$.

c) We say that an element $x$ of G is $p$-regular if its order is prime to $p$; we denote by $G^{reg}$ the set of $p$-regular elements of G and by $\mathscr{Z}_F(G^{reg})$ the space of central functions from $G^{reg}$ to F. From the homomorphism $R_F(G)\rightarrow \mathscr{Z}_F(G^{reg})$ that sends the class of a representation $\pi$ to the restriction of $\chi_{\pi}$ to $G^{reg}$, we deduce a ring homomorphism $\psi : F\otimes_{\mathbf{Z}}R_F(G)\rightarrow \mathscr{Z}_F(G^{reg})$. Prove that $\psi$ is injective (use b) and the corollary of VIII, p. 384).

d) Let $x$ be a $p$-regular element of $G, Z(x)$ its commutant, P a Sylow $p$-subgroup of $Z(x)$, and $H_x$ the subgroup of G generated by P and $x$. Prove that $H_x$ is the direct product of P and the subgroup of G generated by $x$ and that its conjugacy class is well determined by the conjugacy class of $x$.

e) Construct representations $\sigma_1, . . . , \sigma_m$ of $H_x$ of degree 1 and elements $a_1, . . . , a_m$ of F such that the function $\sum a_i\chi_{\sigma_i}$ takes value 1 at $x$ and 0 at the other powers of $x$ (first construct representations of the subgroup generated by $x$). Let $\rho_i=$ Ind$^G_{H_x}\pi_i$ for $i= 1, . . . , m$, and let $f=\sum_ia_i\chi_{\rho_i}$. Prove that $f(x)$ is equal to $(Z(x) : H_x) 1_F$, which is nonzero in F, and that $f(y)$ is zero for every $p$-regular element of G that is not conjugate to $x$.

f ) Deduce from e) that the homomorphism $\psi : F\otimes_{\mathbf{Z}}R_F(G)\rightarrow \mathscr{Z}_F(G^{reg})$ is bijective. In particular, the number of isomorphism classes of simple F[G]-modules is equal to the number of conjugacy classes of $p$-regular elements of G.