Reduced free group

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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 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 and a verbal subgroup of such that
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 .
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.

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 |

Metaproperties

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