variety of groups
Nilpotent groups of class less than form a variety defined by
Similarly, solvable groups of length less than 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.
The largest variety is the variety of all groups, given by an empty set of relations.
By a theorem of Birkhoff, 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 is in , and all subgroups and homomorphic images of members of are also in ).
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 , the free groups are called Burnside groups, and are commonly denoted by , where is the number of generators.
|Title||variety of groups|
|Date of creation||2013-03-22 13:12:02|
|Last modified on||2013-03-22 13:12:02|
|Last modified by||yark (2760)|