# Minimum size of generating set of direct product of two groups may be strictly less than sum of minimum size of generating set of each factor

From Groupprops

## Statement

It is possible to have finite groups such that the minimum size of generating set for is strictly less than the sum of the minimum size of generating set for and for .

## Related facts

- Minimum size of generating set of direct product of two groups is bounded by sum of minimum size of generating set of each factor
- Minimum size of generating set of direct product of groups of coprime order equals maximum of minimum size of generating set of each factor

## Proof

Minimum size of generating set of | Minimum size of generating set of | Minimum size of generating set of | |||
---|---|---|---|---|---|

cyclic group:Z3 | 1 | cyclic group:Z2 | 1 | cyclic group:Z6 | 1 |

symmetric group:S3 | 2 | cyclic group:Z2 | 1 | dihedral group:D12 | 2 |

symmetric group:S3 | 2 | symmetric group:S3 | 2 | direct product of S3 and S3 | 2 |