variety of groups


closed underPlanetmathPlanetmath \PMlinkescapephrasefree groupMathworldPlanetmath


A varietyMathworldPlanetmathPlanetmath of groups is the class of groups G that satisfy a given set of equationally defined relationsMathworldPlanetmathPlanetmathPlanetmath


for all elements x1,x2,x3, of G, where I is some index setMathworldPlanetmathPlanetmath.


Abelian groupsMathworldPlanetmath are a variety defined by the equations


where [x,y]=xyx-1y-1.

Nilpotent groupsMathworldPlanetmath of class less than c form a variety defined by


Similarly, solvable groupsMathworldPlanetmath of length less than c form a variety. (Abelian groups are a special case of both of these.) Note, however, that the class of all nilpotent groups is not a variety, nor is the class of all solvable groups.

For any positive integer n, the variety defined by {x1n=1} consists of all groups of finite exponent dividing n. For n=1 this gives the variety containing only the trivial groups, which is the smallest variety.

The largest variety is the variety of all groups, given by an empty setMathworldPlanetmath of relations.


By a theorem of Birkhoff[1], a class of groups is a variety if and only if it is closed under taking subgroupsMathworldPlanetmathPlanetmath, homomorphic imagesPlanetmathPlanetmathPlanetmath and unrestricted direct products (that is, every unrestricted direct product of members of the class 𝒞 is in 𝒞, and all subgroups and homomorphic images of members of 𝒞 are also in 𝒞).

A variety of groups is a full subcategory of the category of groups, and there is a free group on any set of elements in the variety, which is the usual free group ( modulo the relations of the variety applied to all elements. This satisfies the usual universal propertyMathworldPlanetmath of the free group on groups in the variety, and is thus adjointPlanetmathPlanetmath ( to the forgetful functorMathworldPlanetmathPlanetmath in the category of sets. In the variety of abelian groups, the free groups are the usual free abelian groupsMathworldPlanetmath. In the variety of groups satisfying xn=1, the free groups are called Burnside groups, and are commonly denoted by B(m,n), where m is the number of generatorsPlanetmathPlanetmathPlanetmath.


Title variety of groups
Canonical name VarietyOfGroups
Date of creation 2013-03-22 13:12:02
Last modified on 2013-03-22 13:12:02
Owner yark (2760)
Last modified by yark (2760)
Numerical id 27
Author yark (2760)
Entry type Definition
Classification msc 20E10
Classification msc 20J15
Synonym variety
Related topic GroupVariety
Related topic EquationalClass
Defines Burnside group