# Subgroup rank of a group

From Groupprops

This article defines an arithmetic function on groups

View other such arithmetic functions

## Definition

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

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.