variety of groups
Definition
Examples
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.
For any positive integer , the variety defined by consists of all groups of finite exponent dividing . For 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 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.
References
- 1 G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc., 31 (1935), 433–454.
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 |