variety of groups
Definition
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 if it is closed under taking subgroups^{}, homomorphic images^{} and unrestricted direct products (that is, every unrestricted direct product of members of the class $\mathcal{C}$ is in $\mathcal{C}$, and all subgroups and homomorphic images of members of $\mathcal{C}$ are also in $\mathcal{C}$).
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
- 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 |