Remarks on Brauer correspondences (Q1802214)

Let \(G\) be a finite group, \(H\) a subgroup of \(G\) and \(F\) an algebraically closed field of prime characteristic \(p\). Correspondences with associate blocks of the group algebra \(FG\) to (certain) blocks \(b\) of \(FH\) have been defined by Brauer and Alperin-Burry. It is known that these correspondences coincide where they are both defined. Among others the author shows: (1) If the Alperin-Burry correspondent of \(b\) is defined and has a defect group in common with \(b\) then the Brauer correspondent of \(b\) is also defined. (2) Suppose that the Alperin-Burry correspondent of \(b\) is defined, has defect group \(D\) and that \(H\) is normal in \(G\). Then the Brauer correspondent of \(b\) is defined if and only if \(Z(D)\subseteq H\). More general correspondences have been considered recently by H. I. Blau and W. Wheeler.