Minimum size of generating set of direct product is bounded by sum of minimum size of generating set of each factor

For two groups
Suppose $$A$$ and $$B$$ are groups, and the minimum size of generating set (i.e., the smallest possible size of a  generating set) of $$A$$ is $$a$$ while the minimum size of generating set of $$B$$ is $$b$$. Then, the minimum size of generating set of the external direct product $$A \times B$$ is at most equal to $$a + b$$.

For multiple groups
Suppose $$G_1, G_2, \dots G_n$$ are groups, and the minimum size of generating set for these groups are $$a_1,a_2,\dots,a_n$$ respectively. Then, the external direct product $$G_1 \times G_2 \times \dots \times G_n$$ has minimum size of generating set at most equal to $$a_1 + a_2 + \dots + a_n$$.

Similar facts for direct products

 * 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
 * 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 direct product of groups of coprime order equals maximum of minimum size of generating set of each factor

Similar facts for extensions, subgroups, and quotients

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