# Hopfianness is not subgroup-closed

From Groupprops

This article gives the statement, and possibly proof, of a group property (i.e., Hopfian group)notsatisfying a group metaproperty (i.e., subgroup-closed group property).

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## Statement

It is possible to have a Hopfian group and a subgroup of that is not Hopfian.

## Related facts

### Stronger facts

### Opposite facts

## Proof

Let be free group:F2 and be the derived subgroup of .

- is Hopfian because finitely generated and free implies Hopfian.
- is not Hopfian: is a free group on a countable number of generators. It admits a surjective endomorphism that is not an automorphism obtained via a surjective map that is not bijective from the freely generating set for to itself.