# Maximum size of minimal generating set

## Definition

Suppose $G$ is a group that has at least one minimal generating set, i.e., a generating set with the property that no proper subset of it generates $G$. The maximum size of minimal generating set for $G$ is defined as the maximum of the sizes of all possible minimal generating sets for $G$.

Note that minimal generating sets are also called irredundant generating sets and hence the maximum size of minimal generating set can also be called the maximum size of irredundant generating set.

## Cases where this is defined and undefined

### Finite groups

The maximum size of minimal generating set is defined for any finite group, and can be bounded by various numbers in terms of the order of the group, as described below at #Relation with other arithmetic functions.

### Infinite, finitely generated groups

In the case that $G$ is infinite but finitely generated, it possesses finite minimal generating sets, and in fact, every minimal generating set is finite (see equivalence of definitions of finitely generated group). However, there may or may not be a finite maximum size of minimal generating set. Examples of both kinds are below:

• For the group of integers, there is no upper bound on the maximum size of a minimal generating set. Specifically, for any positive integer $n$, we can take $n+1$ distinct primes, and consider a generating set that comprises products of these primes, $n$ at a time. This is a minimal generating set.
• There do exist infinite groups where there is a maximum size of minimal generating set. For instance, the Tarski groups are infinite groups with the property that any two elements that are not redundant with each other generate the whole group.

## Relation with other arithmetic functions

### Arithmetic functions taking values less than or equal to this

Arithmetic function Meaning Proof of comparison
minimum size of generating set smallest possible size among all generating sets follows from the fact that any generating set of minimum size must of necessity be a minimal generating set.

### Arithmetic functions taking values greater than or equal to this

Arithmetic function Meaning Proof of comparison
max-length of a group maximum possible length of a chain of subgroups of the group size of minimal generating set is bounded by max-length
sum of exponents on prime divisors in a prime factorization of the order (via max-length)