Verbally complete group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
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 a verbally complete group if every word map corresponds to a word that does not reduce (in the free group sense) to the identity element is surjective. In other words, for any word $w(x_{1},x_{2},\dots ,x_{n})$ and any element $g\in G$, there exist solutions in $G$ to $w(x_{1},x_{2},\dots ,x_{n})=g$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
algebraically closed group				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
divisible group	every element has a $n^{th}$ root for every $n$.			|FULL LIST, MORE INFO
group in which every element is a commutator				|FULL LIST, MORE INFO
perfect group	every element is a product of commutators			|FULL LIST, MORE INFO