# variety of groups

## Definition

A variety of groups is the class of groups $G$ that satisfy a given set of equationally defined relations

 $\{\,r_{i}(x_{1},\ldots,x_{n_{i}})=1\mid i\in I\,\}$

for all elements $x_{1},x_{2},x_{3},\ldots$ of $G$, where $I$ is some index set.

## Examples

Abelian groups are a variety defined by the equations

 $\{[x_{1},x_{2}]=1\},$

where $[x,y]=xyx^{-1}y^{-1}$.

Nilpotent groups of class less than $c$ form a variety defined by

 $\{[[\cdots[[x_{1},x_{2}],x_{3}]\cdots],x_{c}]=1\}.$

Similarly, solvable groups 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 $\{x_{1}^{n}=1\}$ 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 set of relations.

## Notes

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

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 (http://planetmath.org/FreeGroup) modulo the relations of the variety applied to all elements. This satisfies the usual universal property of the free group on groups in the variety, and is thus adjoint (http://planetmath.org/AdjointFunctor) to the forgetful functor in the category of sets. In the variety of abelian groups, the free groups are the usual free abelian groups. In the variety of groups satisfying $x^{n}=1$, the free groups are called Burnside groups, and are commonly denoted by $B(m,n)$, where $m$ is the number of generators.

## References

Title variety of groups VarietyOfGroups 2013-03-22 13:12:02 2013-03-22 13:12:02 yark (2760) yark (2760) 27 yark (2760) Definition msc 20E10 msc 20J15 variety GroupVariety EquationalClass Burnside group