Brauer's fixed-point-formula as a consequence of Thompson's order- formula (Q584396)

Ever since its first appearance in the literature, the fixed-point- formula of Richard Brauer has been an extremely useful and handy tool in the study of the local structure of a finite group of even order. H. Wielandt puts Brauer's formula into a more general context, but what one actually needs is to handle situations in which a four-group acts on a group of odd order. In this note, we show that the fixed-point-formula of R. Brauer can be derived from J. G. Thompson's order formula by easy arguments, and in this way we present an alternative proof of Brauer's result. It is elementary in the sense that no character-theoretic argument is involved in it.