# Finitely generated and free implies Hopfian

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 free group) must also satisfy the second group property (i.e., Hopfian group)

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Get more facts about finitely generated free group|Get more facts about Hopfian group

## Contents

## Statement

Any finitely generated free group is Hopfian: it is *not* isomorphic to any proper quotient of itself.

## Related facts

## Facts used

## Proof

The proof follows directly by combining facts (1) and (2).