Linear Methods in Nilpotent Groups

  • Bertram Huppert
  • Norman Blackburn
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 242)


The subject of this chapter is commutator calculation. It will be recalled that the commutator [x, y] of two elements x, y of a group is defined by the relation
$$ [x,y] = {{x}^{{ - 1}}}{{y}^{{ - 1}}}xy. $$
. We then have
$$ [xy,z] = {{[x,z]}^{y}}[y,z],\quad [x,yz] = [x,z]{{[x,y]}^{z}}. $$
. These relations are rather similar to the conditions for bilinearity of forms, and there are a number of ways of formalizing this similarity. Once this is done, commutator calculations can be done by linear methods. Several examples of theorems proved by this method will be given in this chapter.


Linear Method Associative Algebra Nilpotent Group Central Series Invariant Subgroup 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  • Bertram Huppert
    • 1
  • Norman Blackburn
    • 2
  1. 1.Mathematisches Institut der UniversitätMainzGermany
  2. 2.Department of MathematicsThe UniversityGB-ManchesterUK

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