Normal Subgroups, Nilpotent and Solvable Lie Groups

  • Joachim HilgertEmail author
  • Karl-Hermann Neeb
Part of the Springer Monographs in Mathematics book series (SMM)


In this chapter, we address structural aspects of Lie groups. Here an important issue is to see that for any closed normal subgroup N of a Lie group G, the quotient G/N carries a canonical Lie group structure, so that we may consider N and G/N as two pieces into which G decomposes. With this information, we then address the canonical factorization of a morphism of Lie groups into a surjective, a bijective, and an injective one. In particular, we describe some tools to calculate fundamental groups of Lie groups and homogeneous spaces.


Normal Subgroup Fundamental Group Semidirect Product Canonical Factorization Universal Covering Group 
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    Hochschild, G., “The Structure of Lie Groups, ” Holden Day, San Francisco, 1965 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of PaderbornPaderbornGermany
  2. 2.Department of MathematicsFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany

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