Decomposition numbers of $p$-solvable groups

Abstract

In the character theory of finite groups one decom- poses each ordinary irreducible character x% of a group into an in- tegral linear combination of p-modular irreducible characters j, X j. The nonnegative integers da are called the p-decompo- sition numbers. Let G be a ^-solvable group whose p-Sylow sub- groups are abelian. If G/0P'P{G) is cyclic the p-decomposition numbers are g 1. This condition is far from necessary as any group G with abelian, normal p-Sylow subgroup P with G/P abelian has p-decomposition numbers gl. A result of Brauer and Nesbitt together with the first result yields the following. A group G has a normal p-complement and abelian p-Sylow subgroups if and only if each irreducible character of G is irreducible as a p-modular character. A group is said to be ^-solvable if each of its composition factors is either a p-group or a p'-group, p a prime number. Denote by Op>iG) and Op'PiG) the largest normal p'-subgroup of a group G and the in- verse image of the largest normal p-subgroup of G/Op>iG) under the natural homomorphism G—>G/Op>iG) respectively. The following proposition is 1.2.3 of (6). If G is a p-solvable group and P is a p- Sylow subgroup of G then the center of P, Z(P)COP'P(G). As a re- sult, if P is abelian PCZ(P)CtV,,(G) and so Cyp(G) =PCV(G). It also follows that (G: Op>p(G)} is relatively prime to p. We assume knowledge of elementary character theory and recall some facts about the relationship between the ordinary and modular representations of a group G. References (l) and (2) will supply the details. Let K be the extension of the rational numbers obtained by adjoining the mth roots of 1 where m is the exponent of G. R denotes the valuation ring of an extension of the p-adic valuation to K in K, (P denotes the maximal ideal of R and K=R/(P a field of character- istic p. If Z is a ^-representation of G, there exists a i£-similar representa- tion Z' such that all the entries of Z'(g), gEG, lie in R. By reducing the entries of Z'(g) modulo 6° we get a X-representation Z of G. This process does not determine Z uniquely, but the composition factors of Z are uniquely determined by Z.