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

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

Statement

Quantitative version

Suppose $G$ is a group, $H$ is a normal subgroup, and $G/H$ is the corresponding quotient group. If $a$ denotes the minimum size of generating set for $H$ and $b$ denotes the minimum size of generating set for $G/H$ , then the minimum size of generating set for $G$ is at most $a+b$ .

Corollary for finitely generated groups

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

Related facts