Inert subgroup

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 $$H$$ &cap; $$K$$ 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).

Weaker properties

 * Compressed subgroup

Metaproperties
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.

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