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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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