# Intersection of finitely many verbal subgroups

From Groupprops

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

## Definition

Suppose is a group and is a subgroup of . We say that is an **intersection of finitely many verbal subgroups** in if there exists a positive integer and verbal subgroups of such that equals the intersection of subgroups .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

verbal subgroup | generated by a set of words | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

existentially bound-word subgroup | |FULL LIST, MORE INFO | |||

fully invariant subgroup | invariant under all endomorphisms | |FULL LIST, MORE INFO | ||

characteristic subgroup | invariant under all automorphisms | Fully invariant subgroup|FULL LIST, MORE INFO | ||

normal subgroup | invariant under all inner automorphisms | Fully invariant subgroup|FULL LIST, MORE INFO |