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Ladders

  • H. E. A. Eddy CampbellEmail author
  • David L. Wehlau
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  • 1.2k Downloads
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 139)

Abstract

We consider a group G with a normal subgroup N. If σG and τN, then τσ=στ′ for some τ′∈N by normality. Therefore for \(f \in \mathbb{F}[V]^{N}\), we have τ⋅(σf)=τσf=στ′⋅f=σf and thus \(\sigma \cdot f \in \mathbb{F}[V]^{N}\). This shows that G acts on \(\mathbb{F}[V]^{N}\). Clearly \((\mathbb{F}[V]^{N})^{G} = \mathbb{F}[V]^{G}\). Since N acts trivially on \(\mathbb{F}[V]^{N}\), in fact, G/N acts on \(\mathbb{F}[V]^{N}\) and \((\mathbb{F}[V]^{N})^{G/N} = \mathbb{F}[V]^{G}\). We have seen this in detail in Lemma 1.10.1.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Sir Howard Douglas Hall, Dept. MathematicsUniversity of New BrunswickFrederictonCanada
  2. 2.Dept. Mathematics & Computer ScienceRoyal Military College of CanadaKingstonCanada

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