Let G be a finite group, p a rational prime number, and β«,ήβ¬ R, F a ''sufficiently large'' p-modular system. This includes the requirement that R is a complete discrete valuation ring of characteristic 0, with field of Ε½ . fractions β«,ήβ¬ such that F s RrJ R is algebraically closed of characteristic < < p. We also require that β«ήβ¬ contain a primitive G th root of unity. As p usual, for a non-zero integer n and a prime number q, the highest power of q dividing n is denoted by n . The hypotheses on F and R imply that β«ήβ¬ q contains pΠ-roots of unity of all orders. Hence we may, and do, suppose that β«ήβ¬ contains all pΠ-roots of unity in β«,ήβ¬ the complex root of unity e 2 i r <G < p , and the rational field β«.ήβ¬
Let X be a finite pΠ-central extension of a section of G. Then the KrullαSchmidt theorem holds for finitely generated RX modules, as R is complete. We may identify complex characters of X with β«-ήβ¬valued characters of X. Whenever we speak of characters from now on, we will be referring to complex characters. As is now customary, we say that an d Ε½ .
< < irreducible character, , of X with p 1 s X has defect d. p p We will sometimes refer to blocks of the group algebra RX simply as '' p-blocks of X.'' Let b be a block of RX. We denote the set of irreducible Ε½ . characters in b by Irr b . We denote the multiplicative identity of b by 1 . b Ε½ . Now let Y be a central p-subgroup of X. We denote by N N X, Y the set of all normal chains of p-subgroups of X of the form s Q s Y -Q -ΠΈΠΈΠΈ -Q ,