# Inert 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 $H$ of a group $G$ is said to be inert if for any subgroup $K$ of $G$, the rank of $H$$K$ is not more than the rank of $K$.

The notion of inert subgroup is typically of interest inside a free group, where all subgroups are free.

### In terms of the transfer-closure operator

The subgroup property of being inert is obtained by applying the transfer-closure operator to the subgroup property of being a rank-dominated subgroup (that is, of having rank not more than that of the whoel group).

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

The proof of transitivity follows directly from the definition. Equivalently, it follows form the factthat applying the transfer-closure operator to any transitive subgroup property again yields a transitive subgroup property.

### Transfer condition

YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View other subgroup properties satisfying the transfer condition

This follows clearly from the fact that it is obtained as a transfer-closure.