Varieties of Groups

  • Bruce Chandler
  • Wilhelm Magnus
Part of the Studies in the History of Mathematics and Physical Sciences book series (HISTORY, volume 9)


Roughly speaking, a variety of groups is a class of groups in which certain relations called laws or rules or identities are universally valid. The most widely investigated variety of groups is that of abelian groups in which the commutative law is universally valid or, in other words, in which for arbitrary elements x1, x2 of the group, the identity
$$ \left( {{x_1},{x_2}} \right) = 1 $$
holds, where, as usual, (x1, x2) is defined as the commutator
$$ x_1^{ - 1}x_2^{ - 1}{x_1}{x_2} $$
of x1 and x2. Similarly, the Burnside variety of exponent e is defined as the class of groups in which, for all elements x, the relation
$$ {x^e} = 1 $$
holds. The metabelian groups of Chapter II.6 can be defined as the variety of groups define4 by the identity
$$ \left( {\left( {{x_1},{x_2}} \right),\left( {{x_3},{x_4}} \right)} \right) = 1, $$
and the nilpotent groups of class c discussed in Chapter II.7 form the variety defined by the identity
$$ \left( {{x_1},{x_2},...,{x_c},{x_{c + 1}}} \right) = 1 $$
where the parentheses ( ) denote the simple (c + l)-fold commutator defined in Chapter II.7.


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Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • Bruce Chandler
    • 1
  • Wilhelm Magnus
    • 2
  1. 1.The College of Staten Island of The City University of New YorkStaten IslandUSA
  2. 2.New RochelleUSA

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