Brauer third main theorem

For notation and definitions, see Brauer first main theorem.

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 $RH$.

See [a1], [a2], and [a3].

References

[a1]	J.L. Alperin, "Local representation theory" , Cambridge Univ. Press (1986)
[a2]	C. Curtis, I. Reiner, "Methods of representation theory" , II , Wiley (1987)
[a3]	H. Nagao, Y. Tsushima, "Representation of finite groups" , Acad. Press (1987)