Verbally complete 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
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