# Minimum size of generating set of quotient group is at most minimum size of generating set of group

From Groupprops

## Statement

Suppose is a group, is a normal subgroup, and is the corresponding quotient group. Then, the minimum size of generating set of is *at most* equal to the minimum size of generating set of .

## Related facts

- Minimum size of generating set of subgroup may be strictly greater than minimum size of generating set of group
- Minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group
- Minimum size of generating set of direct product is bounded by sum of minimum size of generating set of each factor
- Minimum size of generating set of direct product is bounded below by maximum of minimum size of generating set of each factor