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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: <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$p$">-solvable group, decomposition numbers, <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$p$">-modular character, ordinary character
Article copyright: © Copyright 1970 American Mathematical Society