# Minimum size of generating set

From Groupprops

This article defines an arithmetic function on groups

View other such arithmetic functions

## Contents

## Definition

Let be a group. The **minimum size of generating set** for , often called the **rank** or **generating set-rank** of , and sometimes denoted or , is defined as the minimum possible size of a generating set for .

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 |