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.

Particular cases

Upper bound on minimum size of generating set	Name of groups satisfying this upper bound
0	trivial group
1	cyclic group
2	2-generated group (examples include symmetric group on a finite set, see symmetric group on a finite set is 2-generated)

Facts

• Minimum size of generating set of subgroup may be more than of whole group
• Minimum size of generating set of quotient is less than or equal to that of whole group
• Cyclicity is subgroup-closed, i.e., the property of minimum size of generating set being at most 1 is closed under taking subgroups.
• Minimum size of generating set of direct product of two groups is bounded by sum of minimum size of generating set for each

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.