Nilcentralized and selfcentralizing Brauer pairs in blocks

Abstract

7.1 As in chapter 3, let G be a finite group, k an algebraically closed field of characteristic p, b a block of G and (P, e) a maximal Brauer (b, G)-pair, and denote by F_(b,G) the corresponding Frobenius P-category defined there. In this chapter, we give alternative descriptions of the F_(b,G)-nilcentralized and the F_(b,G)-selfcentralizing subgroups Q of P in terms of the blocks in their associated Brauer (b, G)-pairs. As a matter of fact, in this context the group F_(b,G)(Q) can be canonically lifted to a k^*-group $\hat F_{(b,G)} (Q)$ (cf. 1.23) and there exists a canonical Dade $\bar N_P (Q)$ -algebra S _Q (cf. 1.20).