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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The regular module problem. I


Authors: T. R. Berger, B. B. Hargraves and C. Shelton
Journal: Trans. Amer. Math. Soc. 333 (1992), 251-274
MSC: Primary 20C15
DOI: https://doi.org/10.1090/S0002-9947-1992-1055566-8
MathSciNet review: 1055566
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Abstract: In the study of induced representations the following problem arises: Let $ H = AG$ be a finite solvable group and $ {\mathbf{k}}$ a field with $ \operatorname{char}{\mathbf{k}}\nmid\; \vert A\vert$. Let $ V$ be an irreducible, faithful, primitive $ {\mathbf{k}}[AG]$-module. Suppose $ H$ contains a normal extraspecial $ r$-subgroup $ R$ with $ Z(R) \leq Z(H)$ and that $ A$ acts faithfully on $ R$. Under what conditions does $ A$ have a regular direct summand in $ V$?

In this paper we consider this question under the hypotheses that $ G = MR$, where $ 1 \ne M$ is normal abelian in $ AM$, $ A$ is nilpotent, $ (\vert A\vert,\vert MR\vert) = (\vert M\vert,\vert R\vert) = 1$ , and $ R/Z(R)$ is a faithful, irreducible $ AM$-module. We show that $ A$ has at least three regular direct summands in $ V$ unless $ \vert A\vert$, $ \exp (M)$, and $ r$ satisfy certain very restrictive conditions.


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DOI: https://doi.org/10.1090/S0002-9947-1992-1055566-8
Article copyright: © Copyright 1992 American Mathematical Society