Compressed subgroup

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed compressed if for any subgroup $K$ of $G$ containing $H$, the rank of $H$ is not more than the rank of $K$.

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.

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 condition
ABOUT 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.