Decomposition numbers of $p$-solvable groups
Author:
Forrest Richen
Journal:
Proc. Amer. Math. Soc. 25 (1970), 100-104
MSC:
Primary 20.40
DOI:
https://doi.org/10.1090/S0002-9939-1970-0254146-2
MathSciNet review:
0254146
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Abstract | References | Similar Articles | Additional Information
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