sentences:  These are the coefficients which occur when the restrictions of ordinary characters of formula_9 (from the given block) to elements of the form "tu", where "u" ranges over elements of order prime to "p" in formula_17, are written as linear combinations of the irreducible Brauer characters of formula_17.The content of the theorem is that it is only necessary to use Brauer characters from blocks of formula_17 which are Brauer correspondents of the chosen block of "G".Brauer's third main theorem states that when "Q" is a "p"-subgroup of the finite group "G",
  and "H" is a subgroup of "G," containing formula_1, and contained in formula_2,
  then the principal block of "H" is the only Brauer correspondent of the principal block of "G" (where the blocks referred to are calculated in characteristic "p").