Parafree 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

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