# Brauer second main theorem

Let be an element of whose order is a power of . The -section of associated to is the set of all elements of whose -part is conjugate to . Brauer second main theorem relates the values of irreducible characters of on the -section associated to to values of characters in certain blocks of .
Suppose that is an irreducible character of (cf. also Character of a group), afforded by the -free right -module , and belonging to the block (cf. also Defect group of a block). Let be a -element of , and let . For all -subgroups of , ; hence is defined for all blocks of . One can organize the block decomposition of as . Let be the projection of on , and let be the projection of on . The restriction of to can be decomposed as . If is the character of and is the character of , then of course for all . Brauer's second main theorem states that for all elements of order prime to , . Thus, the values of on the -section associated to are determined in the blocks of sent to by the Brauer correspondence (cf. also Brauer first main theorem).