# Brauer second main theorem

For notation and definitions, see Brauer first 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).

This theorem was first proved in [a1]. See also [a2], [a3], and [a4].

#### References

[a1] | R. Brauer, "Zur Darstellungstheorie der Gruppen endlicher Ordnung II" Math. Z. , 72 (1959) pp. 22–46 |

[a2] | C. Curtis, I. Reiner, "Methods of representation theory" , II , Wiley (1987) |

[a3] | W. Feit, "The representation theory of finite groups" , North-Holland (1982) |

[a4] | H. Nagao, Y. Tsushima, "Representation of finite groups" , Acad. Press (1987) |

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Brauer second main theorem.

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