Minimum size of generating set

This article defines an arithmetic function on groups
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Definition

Let $G$ be a group. The minimum size of generating set for $G$, often called the rank or generating set-rank of $G$, and sometimes denoted $d(G)$ or $r(G)$, is defined as the minimum possible size of a generating set for $G$.

This number is finite if and only if the group is a finitely generated group.

Related notions

• Subgroup rank of a group: This is the maximum of the generating set-ranks over all subgroups of the group.
• Rank of a p-group: For a group of prime power order, this is the maximum of the ranks of all the abelian subgroups of the group.