# Finitely generated not implies residually finite

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., finitely generated group) neednotsatisfy the second group property (i.e., residually finite group)

View a complete list of group property non-implications | View a complete list of group property implications

Get more facts about finitely generated group|Get more facts about residually finite group

## Statement

A finitely generated group need not be a residually finite group. In particular, the notion of finitely generated residually finite group is strictly stronger than the notion of finitely generated group.

## Related facts

- Finitely generated abelian implies residually finite
- Free implies residually finite
- Finitely generated and residually finite implies Hopfian

## Proof

### Any finitely generated simple group

Consider any infinite finitely generated simple group. Since this has no proper nontrivial normal subgroup, it clearly does not have any normal subgroup of finite index, hence it cannot be residually finite. On the other hand, by construction, it is finitely generated.