# Residually finite not implies finitely generated

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., residually finite group) neednotsatisfy the second group property (i.e., finitely generated group)

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## Statement

A residually finite group need not be a finitely generated group. In particular, the property of being a Finitely generated residually finite group (?) is *strictly* stronger than the property of being a residually finite group.

## Related facts

- Finitely generated not implies residually finite
- Finitely generated abelian implies residually finite
- Finitely generated and residually finite implies Hopfian
- Residually finite not implies Hopfian

## Proof

Consider an external direct product (or a restricted external direct product) of infinitely many nontrivial finite groups. In neither case is it finitely generated -- in the unrestricted case because it is uncountable, and in the restricted case because it is a locally finite group that is infinite. However, in both cases, the group is residually finite, using the definition of residually finite as being a subdirect product of finite groups.