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DETERMINING ASCHBACHER CLASSES USING CHARACTERS
Published online by Cambridge University Press: 11 November 2014
Abstract
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Let 
${\rm\Delta}:G\rightarrow \text{GL}(n,K)$ be an absolutely irreducible representation of an arbitrary group 
$G$ over an arbitrary field 
$K$; let 
${\it\chi}:G\rightarrow K:g\mapsto \text{tr}({\rm\Delta}(g))$ be its character. In this paper, we assume knowledge of 
${\it\chi}$ only, and study which properties of 
${\rm\Delta}$ can be inferred. We prove criteria to decide whether 
${\rm\Delta}$ preserves a form, is realizable over a subfield, or acts imprimitively on 
$K^{n\times 1}$. If 
$K$ is finite, we can decide whether the image of 
${\rm\Delta}$ belongs to certain Aschbacher classes.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © 2014 Australian Mathematical Publishing Association Inc.
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