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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Decomposition numbers of $ p$-solvable groups

Author: Forrest Richen
Journal: Proc. Amer. Math. Soc. 25 (1970), 100-104
MSC: Primary 20.40
MathSciNet review: 0254146
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Abstract: In the character theory of finite groups one decomposes each ordinary irreducible character $ {\chi _i}$ of a group into an integral linear combination of $ p$-modular irreducible characters $ {\phi _j},{\chi _i} = \sum {{d_{ij}}{\phi _j}} $. The nonnegative integers $ {d_{ij}}$ are called the $ p$-decomposition numbers. Let $ G$ be a $ p$-solvable group whose $ p$-Sylow subgroups are abelian. If $ G/{O_{p'p}}(G)$ is cyclic the $ p$-decomposition numbers are $ \leqq 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 $ \leqq 1$. 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.

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Keywords: $ p$-solvable group, decomposition numbers, $ p$-modular character, ordinary character
Article copyright: © Copyright 1970 American Mathematical Society