Verbally closed subgroup

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]

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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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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
retract				|FULL LIST, MORE INFO
direct factor		(via retract)	(via retract)	|FULL LIST, MORE INFO

Weaker 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)	|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)	|FULL LIST, MORE INFO
powering-invariant subgroup		(via divisibility-closed)	(via divisibility-closed)	|FULL LIST, MORE INFO