# Minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., finitely generated group) satisfying a group metaproperty (i.e., extension-closed group property)

View all group metaproperty satisfactions | View all group metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for group properties

Get more facts about finitely generated group |Get facts that use property satisfaction of finitely generated group | Get facts that use property satisfaction of finitely generated group|Get more facts about extension-closed group property

## Contents

## Statement

### Quantitative version

Suppose is a group, is a normal subgroup, and is the corresponding quotient group. If denotes the minimum size of generating set for and denotes the minimum size of generating set for , then the minimum size of generating set for is *at most* .

### Corollary for finitely generated groups

Suppose is a group, is a normal subgroup, and is the corresponding quotient group. Then, if both and are finitely generated groups, so is .

## 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 quotient group is at most minimum size of generating set of 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
- Minimum size of generating set of group generated by subgroups is bounded by sum of minimum size of generating set of each subgroup
- Minimum size of generating set of direct product of groups of coprime order equals maximum of minimum size of generating set of each factor