Parafree 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

A group G is termed parafree (sometimes also absolutely parafree to distinguish from the relative notion thati s useful when dealing with subvarieties of the variety of groups) if it satisfies both these conditions:

  1. It is a residually nilpotent group.
  2. There exists a free group F and a homomorphism of groups that induces identity maps on each quotient group between successive members of the lower central seriesExplicitly, for every positive integer i, \varphi induces an isomorphism \gamma_i(F)/\gamma_{i+1}(F) \cong \gamma_i(G)/\gamma_{i+1}(G).

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
free group