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

From Groupprops

This article gives an upper 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 above by the sum of the values for the direct factors.

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## Contents

## Statement

### For two groups

Suppose and are groups, and the minimum size of generating set (i.e., the smallest possible size of a generating set) of is while the minimum size of generating set of is . Then, the minimum size of generating set of the external direct product is at *most* equal to .

### For multiple groups

Suppose are groups, and the minimum size of generating set for these groups are respectively. Then, the external direct product has minimum size of generating set at most equal to .

## Related facts

### 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