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]
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
retract |FULL LIST, MORE INFO
direct factor (via retract) (via retract) 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) 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
powering-invariant subgroup (via divisibility-closed) (via divisibility-closed) Local divisibility-closed subgroup|FULL LIST, MORE INFO