Subgroup rank of a group

Definition
Suppose $$G$$ is a group. Then, the subgroup rank of $$G$$ is defined as the supremum, over all subgroups $$H$$ of $$G$$, of the minimum size of generating set of $$H$$.

If the subgroup rank of a group is finite, then the group is a slender group, i.e., every subgroup of it is a finitely generated group.

Related notions

 * Minimum size of generating set