# 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 |
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 $d$, for some pre-specified $d$ |
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 |