Reduced free group

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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Definition

No.	Shorthand	A group is termed a reduced free group if ...	A group $G$ is termed a reduced free group if ...
1	Quotient of free	it is isomorphic to the quotient of a free group by a verbal subgroup	there is a free group $F$ and a verbal subgroup $V$ of $F$ such that $G \cong F/V$
2	Free in some subvariety	it is a free algebra in some subvariety of the variety of groups.	there is a subvariety $\mathcal{V}$ of the variety of groups such that $G$ is a free algebra in that subvariety. More explicitly, there is a generating set $S$ for $G$ such that for any $H \in \mathcal{V}$, any set map $f:S \to H$ extends uniquely to a group homomorphism from $G$ to $H$.
3	Free in own subvariety	it is a free algebra in the subvariety of the variety of groups generated by itself.	$G$ is a free algebra in the subvariety $\mathcal{V}(G)$ (i.e., the subvariety generated by $G$) in the variety of groups.

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			|FULL LIST, MORE INFO
Free abelian group	free in the variety of abelian groups			|FULL LIST, MORE INFO
Elementary abelian group	trivial or abelian of prime exponent			|FULL LIST, MORE INFO
Burnside group	free in the variety of groups of exponent dividing $d$, for some pre-specified $d$			|FULL LIST, MORE INFO
Finite homocyclic group	finite direct power of a finite cyclic group			|FULL LIST, MORE INFO
Free class two group	free in the variety of groups of nilpotency class two			|FULL LIST, MORE INFO
Free metabelian group	free in the variety of metabelian groups			|FULL LIST, MORE INFO

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	|FULL LIST, MORE INFO

Metaproperties

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