Explicit Brauer induction and the Glauberman correspondence
Let S and G be finite groups of coprime order such that S acts on G. If S is solvable, Glauberman  proves the existence of a bijection between the S-fixed irreducible representations of G and the irreducible representations of Gs. In the case of G solvable, Isaacs  uses a totally different method to prove the existence of a bijection between the same two sets of representations. Assuming the existence of the Glauberman correspondence, Boltje  uses the method of Explicit Brauer Induction (EBI) to give an explicit version of this correspondence for the case in which S is a p-group. After presenting the above results, we outline a strategy for investigating these correspondences using Explicit Brauer Induction, and we use these ideas to give a new proof for the theorems of Glauberman and Boltje. We move on to suggest some ideas of how this work may extend to Isaacs' correspondence. We also mention a link to Shintani's correspondence . In the final chapter, we look at cryptography, and mention a potential application of some of our techniques (Adams Operations) in this field.