Subgroup rank of a group

This article defines an arithmetic function on groups
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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