Minimum size of generating set

This article defines an arithmetic function on groups
View other such arithmetic functions

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

Relation with other arithmetic functions

Arithmetic functions defined using it

• 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.

Arithmetic functions taking values greater than or equal to minimum size of generating set

Arithmetic function	Meaning	Proof of comparison
maximum size of minimal generating set	The maximum, over all minimal generating sets of the group, of their sizes
max-length of a group	maximum possible length of a subgroup series for the group	(via maximum size of minimal generating set)
sum of exponents of prime divisors in prime factorization of order		(via max-length). See minimum size of generating set is bounded by sum of exponents of prime divisors of order