# 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
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## 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
divisible group every element has a $n^{th}$ root for every $n$. |FULL LIST, MORE INFO