# Minimum size of generating set of direct product is bounded below by maximum of minimum size of generating set of each factor

This article gives a lower bound on the value of the arithmetic function minimum size of generating set of an in terms of the values for the direct factors. It says that the value for the direct product is bounded from below by the maximum of the values for the direct factors.
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## Statement

### 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 least equal to $\max \{ a, b \}$.

### 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 least equal to $\max \{ a_1, a_2, \dots, a_n \}$.