sentences:  Brauer's first main theorem states that if formula_9 is a finite group a formula_10 is a formula_11-subgroup of formula_9, then there is a bijection between the set of 
  (characteristic "p") blocks of formula_9 with defect group formula_10 and blocks of the normalizer formula_15 with 
  defect group "D".This bijection arises because when formula_16, each block of "G"
  with defect group "D" has a unique Brauer correspondent block of "H", which also has defect 
  group "D".Brauer's second main theorem gives, for an element "t" whose order is a power of a prime "p", a criterion for a (characteristic "p") block of formula_17 to correspond to a given block of formula_9, via "generalized decomposition numbers".