Reduced free group
From Groupprops
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Contents
Definition
| No. | Shorthand | A group is termed a reduced free group if ... | A group  is termed a reduced free group if ...
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|---|---|---|---|
| 1 | Quotient of free | it is isomorphic to the quotient of a free group by a verbal subgroup | there is a free group  and a verbal subgroup  of such that 
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| 2 | Free in some subvariety | it is a free algebra in some subvariety of the variety of groups. | there is a subvariety  of the variety of groups such that is a free algebra in that subvariety. More explicitly, there is a generating set  for such that for any  , any set map  extends uniquely to a group homomorphism from to  .
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| 3 | Free in own subvariety | it is a free algebra in the subvariety of the variety of groups generated by itself. | is a free algebra in the subvariety  (i.e., the subvariety generated by ) in the variety of groups.
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Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Free group | free in the variety of groups | | | ||
| Free abelian group | free in the variety of abelian groups | | | ||
| Elementary abelian group | trivial or abelian of prime exponent | | | ||
| Burnside group | free in the variety of groups of exponent dividing  , for some pre-specified |
| | ||
| Finite homocyclic group | finite direct power of a finite cyclic group | | | ||
| Free class two group | free in the variety of groups of nilpotency class two | | | ||
| Free metabelian group | free in the variety of metabelian groups | | |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Group in which every fully invariant subgroup is verbal | every fully invariant subgroup is a verbal subgroup | fully invariant implies verbal in reduced free | simple groups give counterexamples | | |
