Archiv der Mathematik

, Volume 100, Issue 1, pp 7–24 | Cite as

On the descent of Levi factors



Let G be a linear algebraic group over a field k of characteristic p > 0, and suppose that the unipotent radical R of G is defined and split over k. By a Levi factor of G, one means a closed subgroup M which is a complement to R in G. In this paper, we give two results related to the descent of Levi factors. First, suppose is a finite Galois extension of k for which the extension degree [ : k] is relatively prime to p. If G / has a Levi decomposition, we show that G has a Levi decomposition. Second, suppose that there is a G-equivariant isomorphism of algebraic groups \({R \simeq Lie(R)}\) – i.e. R is a vector group with a linear action of the reductive quotient G/R. If \({G_{{/k}_{sep}}}\) has a Levi decomposition for a separable closure k sep of k, then G has a Levi decomposition. Finally, we give an example of a disconnected, abelian, linear algebraic group G for which \({G_{{/k}_{sep}}}\) has a Levi decomposition, but G itself has no Levi decomposition.

Mathematics Subject Classification (2010)

20G15 20G10 


Linear algebraic group Positive characteristic 


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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of MathematicsTufts UniversityMedfordUSA

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