# Compressed 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]

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

## Definition

### Definition with symbols

A subgroup of a group is termed **compressed** if for any subgroup of containing , the rank of is not more than the rank of .

We typically study compressed subgroups inside a free group.

### In terms of the intermediately operator

The subgroup property of being compressed is obtained by applying the intermediately operator to the subgroup property of being a rank-dominated subgroup. A subgroup is said to be rank-dominated in a group if the rank of the subgroup is not more than the rank of the group.

## Relation with other properties

### Stronger properties

### Weaker properties

- Finite-index-maximal subgroup in case we are working within a free group

## Metaproperties

### Transitivity

It is not clear whether any compressed subgroup of a compressed subgroup is compressed. The problem lies with the fact that intermediate subgroups containing the smaller subgroup may not be comparable either way with the bigger subgroup.

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If a subgroup is compressed in terms of the whole group, it is clearly compressed in every intermediate subgroup.

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Clearly, the trivial subgroup is compressed in any group, because it is generated by zero elements. Also, every group is compressed as a subgroup of itself.