# Verbally closed subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

Suppose $G$ is a group and $H$ is a subgroup of $G$. We say that $H$ is verbally closed in $G$ if the following is true: For any word $w$ in $n$ letters, the image of $H^n$ under the word map corresponding to $w$ equals the intersection of $H$ with the image of $G^n$ under the word map corresponding to $w$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
local divisibility-closed subgroup if an element of the subgroup has a $n^{th}$ root in the whole group, there is a $n^{th}$ root in the subgroup. |FULL LIST, MORE INFO
divisibility-closed subgroup if every element of the group has a $n^{th}$ root, every element of the subgroup has a $n^{th}$ root in the subgroup. (via local divisibility-closed) (via local divisibility-closed) Local divisibility-closed subgroup|FULL LIST, MORE INFO
local powering-invariant subgroup if an element of the subgroup has a unique $n^{th}$ root in the group, that root is in the subgroup. (via local divisibility-closed) (via local divisibility-closed) Local divisibility-closed subgroup|FULL LIST, MORE INFO