Decomposition numbers of 𝑝-solvable groups

Abstract

In the character theory of finite groups one decomposes each ordinary irreducible character χ i {\chi _i} of a group into an integral linear combination of p p -modular irreducible characters ϕ j , χ i = ∑ d i j ϕ j {\phi _j},{\chi _i} = \sum {{d_{ij}}{\phi _j}} . The nonnegative integers d i j {d_{ij}} are called the p p -decomposition numbers. Let G G be a p p -solvable group whose p p -Sylow subgroups are abelian. If G / O p ′ p ( G ) G/{O_{p’p}}(G) is cyclic the p p -decomposition numbers are ≦ 1 \leqq 1 . This condition is far from necessary as any group G G with abelian, normal p p -Sylow subgroup P P with G / P G/P abelian has p p -decomposition numbers ≦ 1 \leqq 1 . A result of Brauer and Nesbitt together with the first result yields the following. A group G G has a normal p p -complement and abelian p p -Sylow subgroups if and only if each irreducible character of G G is irreducible as a p p -modular character.