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Transitive Actions

  • A. L. Onishchik
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 20)

Abstract

1.1. Definitions and Examples. Recall that any transitive action of a group G on a set M is isomorphic to an action of this group by left translations on the set of all cosets G/H where H = G x is the stabilizer of any point xM. In the differentiable case (see 1.4 of Chap. 1) G/H is an analytic homogeneous space of the Lie group G and the isomorphism is a diffeomorphism. The G- space G/H is called a group model (or a Klein model) of the homogeneous space M. A group model depends on the choice of the point x so that the subgroup H is defined up to conjugation in G. By means of a group model any property of the homogeneous space M can be expressed in terms of the group G and a subgroup H. We shall now illustrate this method on the simplest examples.

Keywords

Homogeneous Space Maximal Compact Subgroup Riemannian Structure Transitive Action Homogeneous Vector Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • A. L. Onishchik
    • 1
  1. 1.Yaroslavl UniversityYaroslavlRussia

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