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

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

 * 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