variety of groups

closed underfree group

Definition

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

$$\{r_i(x_1,\mathrm{\dots },x_{n_i})=1\mid i\in I\}$$

for all elements $x_1,x_2,x_3,\mathrm{\dots }$ of $G$,where $I$ is some index set.

Examples

Abelian groups are a variety defined by the equations

$$\{[x_1,x_2]=1\},$$

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

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

$$\{[[\mathrm{\cdots }[[x_1,x_2],x_3]\mathrm{\cdots }],x_c]=1\}.$$

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 ifit is closed under taking subgroups, homomorphic imagesand unrestricted direct products(that is,every unrestricted direct product of membersof 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 (http://planetmath.org/FreeGroup)modulo the relations of the variety applied to all elements.This satisfies the usual universal propertyof the free group on groups in the variety,and is thus adjoint (http://planetmath.org/AdjointFunctor) to the forgetful functorin 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.