Brauer's main theorems

Statement

Let $G$ be a finite group, let $p$ be a prime, and fix a $p$-modular system with residue field $k$ of characteristic $p$. The $p$-blocks of a finite group $H$ are the blocks of the group algebra $k[H]$. Each irreducible complex character and each irreducible Brauer character of $H$ lies in exactly one block. Each block $b$ has a defect group, a $p$-subgroup of $H$ determined up to conjugacy. Each block $b$ of $H$ has a central character $\lambda_b \colon Z(k[H]) \to k$. For a subset $X$ of $G$, let $\hat{X}$ be the sum of the elements of $X$ in $k[G]$. Let $H \le G$ and let $b$ be a block of $H$. Suppose there is a block $B$ of $G$ with $\lambda_B(\hat{C}) = \lambda_b(\widehat{C \cap H})$ for each conjugacy class $C$ of $G$. Then $B$ is unique, one says that $b^G$ is defined, and $b^G = B$ is the Brauer correspondent of $b$. The principal block is the block containing the trivial character.

1. First main theorem. Let $D$ be a $p$-subgroup of $G$ with normaliser $N_G(D)$. The map $b \mapsto b^G$ is a bijection from the blocks of $N_G(D)$ with defect group $D$ onto the blocks of $G$ with defect group $D$.
2. Second main theorem. Let $x \in G$ have order a power of $p$, and let $H = C_G(x)$ be its centraliser. Let $\chi$ be an irreducible complex character of $G$. There are unique complex numbers $d^x_{\chi\varphi}$, for $\varphi$ in the set of irreducible Brauer characters of $H$, such that $\chi(xy) = \sum_{\varphi} d^x_{\chi\varphi} \, \varphi(y)$ for every $p$-regular $y \in H$. Let $\chi$ lie in the block $B$ and $\varphi$ in the block $b$ of $H$. If $d^x_{\chi\varphi} \ne 0$, then $b^G = B$.
3. Third main theorem. Let $H \le G$ and let $b$ be a block of $H$ with defect group $D$ such that $C_G(D) \le H$. Then $b^G$ is defined, and $b^G$ is the principal block of $G$ if and only if $b$ is the principal block of $H$.