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

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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 \}.

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