# Minimum size of generating set

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)

## Relation with other arithmetic functions

### 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