Lanne's \textbf{T}-functor and Hypersurfaces
Received:September 17, 2016  Revised:September 01, 2017
Key Word: \textbf{T}-functor   hypersurface   pointwise stabilizers  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11371343).
Author NameAffiliation
Wenhua ZHENG School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China 
Jizhu NAN School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China 
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Abstract:
      Through discussing the transformation of the invariant ideals, we firstly prove that the \textbf{T}-functor can only decrease the embedding dimension in the category of unstable algebras over the Steenrod algebra. As a corollary we obtain that the \textbf{T}-functor preserves the hypersurfaces in the category of unstable algebras. Then with the applications of these results to invariant theory, we provide an alternative proof that if the invariant of a finite group is a hypersurface, then so are its stabilizer subgroups. Moreover, by several counter-examples we demonstrate that if the invariants of the stabilizer subgroups or Sylow $p$-subgroups are hypersurfaces, the invariant of the group itself is not necessarily a hypersurface.
Citation:
DOI:10.3770/j.issn:2095-2651.2018.01.008
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