# Finitely generated abelian implies residually finite

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finitely generated abelian group) must also satisfy the second group property (i.e., residually finite group)

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## Contents

## Statement

Any finitely generated abelian group (i.e., an abelian group that is also a finitely generated group) is a residually finite group -- for every non-identity element, there is a normal subgroup of finite index not containing it.

## Related facts

- Finitely generated not implies residually finite
- Residually finite not implies finitely generated
- Free implies residually finite
- Free abelian implies residually finite
- Residual finiteness is direct product-closed

## Facts used

- Structure theorem for finitely generated abelian group
- The infinite cyclic group is residually finite.
- Residual finiteness is direct product-closed

## Proof

### Hands-on proof

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### Proof using given facts

By fact (1), any finitely generated abelian group is a direct product of copies of the infinite cyclic group and a finite group. The infinite cyclic group is residually finite (fact (2)) and the finite group is residually finite, so by fact (3), the whole group is residually finite.