# Brauer third main theorem

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Brauer's third main theorem deals with one situation in which the Brauer correspondence (cf. also Brauer first main theorem) is easy to compute. The principal character of a group $G$ is defined to be the character $\chi$ such that $\chi(g)=1$ for all $g\in G$ (cf. also Character of a group). The block to which it belongs is called the principal block of the group algebra $RG$. The defect groups (cf. also Defect group of a block) of the principal block are the Sylow $p$-subgroups of $G$ (cf. also $p$-group). Let $H$ be a subgroup of $G$, and let $b$ be a block of $H$ with defect group $D$ such that $C_G(D)\subseteq H$. Brauer's third main theorem states that $b^G$ is the principal block of $RG$ if and only if $b$ is the principal block of $RH$.