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variety of groups
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(Definition)
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A variety of groups is the class of groups  that satisfy a given set of equationally defined relations
for all elements
 of  , where  is some index set.
Abelian groups are a variety defined by the equations
where
![$ [x,y]=xyx^{-1}y^{-1}$](http://images.planetmath.org:8080/cache/objects/3662/l2h/img7.png "$ [x,y]=xyx^{-1}y^{-1}$") .
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.
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 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 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.
- 1
- G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc., 31 (1935), 433-454.
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"variety of groups" is owned by yark. [ full author list (3) | owner history (2) ]
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(view preamble)
Cross-references: generators, number, free abelian groups, free groups, category of sets, forgetful functor, universal property, category, full subcategory, unrestricted direct products, homomorphic images, subgroups, empty set, exponent, finite, integer, positive, solvable groups, nilpotent groups, equations, abelian groups, index set, relations, class, groups
There are 9 references to this entry.
This is version 24 of variety of groups, born on 2002-12-05, modified 2007-07-25.
Object id is 3662, canonical name is VarietyOfGroups.
Accessed 3625 times total.
Classification:
| AMS MSC: | 20E10 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Quasivarieties and varieties of groups) | | | 20J15 (Group theory and generalizations :: Connections with homological algebra and category theory :: Category of groups) |
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Pending Errata and Addenda
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![$\displaystyle \{[x_1,x_2]=1\},$](http://images.planetmath.org:8080/cache/objects/3662/l2h/img6.png "$\displaystyle \{[x_1,x_2]=1\},$"){kind=small}
![$\displaystyle \{[[\cdots[[x_1,x_2],x_3]\cdots],x_c]=1\}.$](http://images.planetmath.org:8080/cache/objects/3662/l2h/img9.png "$\displaystyle \{[[\cdots[[x_1,x_2],x_3]\cdots],x_c]=1\}.$"){kind=small}