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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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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