Dade's Conjecture for the McLaughlin Simple Group

Abstract

In (Da92) E.C. Dade announcing the first of what has to be a series of conjectures concerned with counting characters in the blocks of finite groups. Specifically, Dade's so-called Ordinary Conjecture asserts that if a finite group G has trivial p-core and B is a p-block of G of positive defect, then the number $k(B,\ d)$ of complex irreducible characters with fixed defect d belonging to B can be expressed as an alternating sum over the corresponding numbers for the normalizers of the non-trivial p-chains of G. Refined versions of this conjecture have been given in subsequent articles. See (Da94) and (DaPr) for further details. Dade claims that the strongest form of his conjectures is true for all finite groups if it is true for all covering groups of finite simple groups. Thus, in order to prove his conjectures, one merely has to go through the complete list of finite simple groups, carefully verifying that the strongest form holds for each one. In this Thesis we prove the most general version of Dade's Conjectures for all covering groups of the McLaughlin group Mc. This is one of the 26 sporadic simple groups. So our Thesis forms a portion of the as yet incomplete proof of Dade's Conjectures for all finite groups.