The History of Combinatorial Group Theory pp 157-161 | Cite as

# Varieties of Groups

Chapter

## Abstract

Roughly speaking, a variety of groups is a class of groups in which certain relations called holds, where, as usual, ( of holds. The metabelian groups of Chapter II.6 can be defined as the variety of groups define4 by the identity and the nilpotent groups of class c discussed in Chapter II.7 form the variety defined by the identity where the parentheses ( ) denote the simple (

*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*x*_{1},*x*_{2}of the group, the identity$$ \left( {{x_1},{x_2}} \right) = 1 $$

*x*_{1},*x*_{2}) is defined as the commutator$$ x_1^{ - 1}x_2^{ - 1}{x_1}{x_2} $$

*x*_{1}and*x*_{2}. Similarly, the Burnside variety of exponent*e*is defined as the class of groups in which, for all elements*x*, the relation$$ {x^e} = 1 $$

$$ \left( {\left( {{x_1},{x_2}} \right),\left( {{x_3},{x_4}} \right)} \right) = 1, $$

$$ \left( {{x_1},{x_2},...,{x_c},{x_{c + 1}}} \right) = 1 $$

*c*+ l)-fold commutator defined in Chapter II.7.## Preview

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## Copyright information

© Springer-Verlag New York Inc. 1982