# 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]

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

## Contents

## Definition

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

## 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 root in the whole group, there is a root in the subgroup. | |FULL LIST, MORE INFO | ||

divisibility-closed subgroup | if every element of the group has a root, every element of the subgroup has a 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 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 |